On Dirichlet, Poncelet and Abel problems

We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary biquadratic algebraic curve is considered. It is shown that a solution is non-unique if and only if a corresponding Poncelet problem for two conics has a periodic trajectory. A set of problems is proven to be equivalent to the above problem. Among them are the solvability problem of the algebraic Pell-Abel equation and the indeterminacy problem of a new moment problem that generalizes the well-known trigonometrical moment problem. Solvability criteria of the above-mentioned problems can be represented in form $\theta\in\Bbb Q$ where number $\theta=m/n$ is built by means of data for a problem to solve. We also demonstrate close relations of the above-mentioned problems to such problems of modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg $XY$-chain and biorthogonal rational functions on elliptic grids in the theory of the Pad\'e interpolation.


Introduction
This work is devoted to new connections between some problems of mathematics such as some ill-posed boundary value problems in a bounded semialgebraic domain for partial differential equations, a moment problem, the Poncelet problem of the projective geometry, the algebraic Pell-Abel equation and some other problems, recently revealed by authors.
Study of ill-posed boundary value problems in bounded domains for partial differential equations go back to J.Hadamard and they are a popular object of modern infestigations (see s. 2.1). In this paper we will examine generally the Dirichlet problem for the string equation The functions u, φ are assumed to be complex-valued functions of real variables. We will consider this and other boundary value problems in semialgebraic domains, the boundary of which is given by so-called biquadratic algebraic curve a ik x i y k = 0 (1.4) where x i , y k are powers. We will consider canonical forms of the curve (1.4) to which the generic curve can be transformed by linear-fractional replacements and we will give criterions of uniqueness breakdown in the form where the number τ is determined by the curve C. Our investigations are based on a study of the John mapping generated on C by characteristics of the equation (1.1), see propositions 29 (s. 6) and 19 (s. 3.5) where we use that this John mapping becomes an usual shift after a transform on universal covering group of the variety (1.4).
We will have observed a remarkable connection of this problem with the Poncelet problem. The Poncelet problem is one of famous problems of projective geometry and it by itself has numerous links with a set of different problems of analysis and physics (see [13], [10] [26] and below s. 4.1). We will prove that for generic biquadratic curve C the Dirichlet problem has a non-unique solution if and only if corresponding the Poncelet problem has a periodic trajectory (proposition 25, s. 4.5). And therefore we will give a new criterion of periodicity in the form (1.5) that differs from the well-known Cayley criterion (see s. 4.6).
Remind that the existence of a periodic trajectory in the Poncelet problem means that each trajectory is periodic by the big Poncelet theorem [13]. Note that different cases of disposition of conics give different cases of curves C and a new setting of the Dirichlet problem (1.1),(1.3) for unbounded domains is more suitable than the classical setting (see s. 2.2).
We will observe one more remarkable connection of the Poncelet problem with the solvability problem of algerbraic the Pell-Abel equation where for given polynomial R(t) of one variable t one should find polynomials P, Q and a constant L satisfying the equation. The Pell-Abel equation solvability is one of famous problems also and this algebraic problem by itself has numerous connections with many problems of analysis (see below s. 5). We will prove (see proposition 27 of s. 5.2) that the Pell-Abel equation (5.1) with a polynomial R of the third or fourth order has a solution if and only if corresponding the Poncelet problem has a periodic trajectory with an even period. On this way we obtain a new criterion in the form (1.5) which differs from wellknown others, for example, from well-known Zolotarev's porcupine (see [43]) . We will show that the same condition (1.5) is a criterion of solution uniqueness to within an additive constant of the Neumann problem u ν * | C = ψ (1.7) for the same equation (1.1) in the same domain where u ν * is a derivative with respect to the conormal ν * (statement 3 of s. 2.3). We will show also that the same condition is a criterion of indeterminacy of a moment problem on the curve C (statement 3 of s. 2.3). By means of the duality equation-domain we will obtain an equivalent problem (2.22) in the form of a hyperbolic (in some cases of corresponding Poncelet problem) equation of the fourth order with only two boundary data on characteristics instead four as in the boundary value problem of the Goursat type, where we will have solution uniqueness almost always. We will have observed some connections with a problem on classical XY-spin chains and a problem from the theory of Toda chain. We will have shown an equivalence of considered problems and also we give an interpretation of this criterion in terms of the John mapping. About links with some other problems of analysis see [58], [41], [43], [51].
Note that some results of present work have published already as short and incomplete fragments in papers [18], [19], [20].
Remark also that some explicit necessary and sufficient conditions of uniqueness breakdown of Dirichlet problem (and some others boundary value problems) solution for partial differential equations with constant coefficients have obtained (see e.g., [16] and [17]) in an arbitrary ellipse. Answers in that works were formulated just in the form of condition (1.5) and prompted answers in our present investigations. In this paper we would like to open a way to examine ill-posed boundary value problems for partial differential equations in some more complicated domains than a circle along with a survey of another fields of mathematics about topics with equivalent contents.
The main goal of the present paper is thus to show numerous relations of the considered problem with remote branches of mathematics and mathematical physics.
The paper is organized as follows.
In Section 2 we describe a theory of uniqueness of Dirichlet problem and the John algorithm. In section 3 we apply this technique to the concrete choice of the biquadratic curve. In Section 4 we consider the solution of the Poncelet problem and show its relation with the John algorithm for the generic biquadratic curve. Section 5 is devoted to the Abel problem of reducing of elliptic integrals to elementary ones. We show how this problem can be formulated in terms of the Poncelet problem (or the John algorithm for the biquadratic curve). In Section 6 we consider relation of our problems with Ritt's problem of existence of periodic functions with a nontrivial multiplication property. Finally, in Section 7 we consider 3 problems in mathematical physics which are related with the Poncelet (or the John algorithm) problem: static solutions of the classical XY Heisenberg chain, elliptic solutions of the Toda chain and elliptic grids for biorthogonal rational functions in the theory of rational Padé interpolation .

Boundary value problems in a domain for the string equation
2.1. Bibliographical remarks. Investigations of ill-posed boundary value problems in bounded domains for partial differential equations go back to J.Hadamard [30] and then A.Huber [32] who for the first time noted nonuniqueness of the solution of the Dirichlet problem for the equation of string vibration (string equation) in a rectangular. Boundary value problems in bounded domains for nonelliptic partial differential equations were regularly studied on the whole in a parallelepiped, besides in domains with a general boundary questions of solution uniqueness of the Dirichlet problem for the hyperbolic equation of the second order on the plane (see reviews and results in [48] were usually studied. In work [14] D.Burgin and R.Duffin have examined the homogeneous Dirichlet problem for the equation u tt − u xx = 0 in a rectangular {0 ≤ t ≤ T ; 0 ≤ x ≤ X}. It is shown that if the ratio T /X is irrational, uniqueness in space of continuously differentiated functions with summable second derivatives takes place. Theorems of existence of solutions in classical spaces are established, and smoothness of the solution is that more than smoothness of boundary function is big and than the number T /X is worse approximated by rational numbers. The Neumann problem is considered also. In works by B.Yo.Ptashnik and his pupils boundary value problems in a parallelepiped for a wide class of the differential equations and systems are investigated, see [48]. All these works (excepting the mentioned work [6]) are based on the methods, essentially using representation of domain as a topological product.
For nonrectangular domains the Dirichlet problem for the string equation was studied in connection with a number of the Denjoy-Poincare rotation (see, e.g., Z.Nitetsky's book [46]) of a homeomorphism of domain boundary, constructed on characteristics of equation (so-called an automorphism of characteristic billiards by Fritz John [34]). A connection of properties of the Dirichlet problem with properties of this homeomorphism has used even in mentioned works by J. Hadamard and A.Huber. To the analysis this connection has undergone in F.John's work [34]. In works by R.A.Alexandrjan and his pupils investigations of this problem and, in particular, of this connection was continued ( [3], [4], [47], [5]). The question on uniqueness of the solution of the Dirichlet problem in this ideology for domains that are convex with respect to characteristics families, should be transformed to a question on irrationality of number of rotation or, that is the same, to a question on presence a continuum set of finite orbits (cycles) of the discrete dynamic system generated by mentioned John homeomorphism. The same questions in connection with an asymptotic behavior of the solution of the Sobolev equation that describe surface oscillations of a fluid filling a body which is flying in atmosphere, were investigated by the siberian mathematicians T.I.Zelenjak, I.V.Fokin and others ( [62], [24]). Researches of the string equation are included in well-known Yu.M.Berezansky's book [12] also, they give possibilities to build domains with angles in which the homogeneous Dirichlet problem is weakly solvable and well-posed concerning the right part and small moves of boundary of the domain, leaving angles in specified limits. Note more that the case of an ellipse was considered in works by A. Huber [32], Alexandrjan [4], V.I.Arnold [6], for small smoothness and for more general equations see the book [17].

John condition.
For the problem (1.1), (1.2) in some general bounded domain Ω Fritz John [34] considered a remarkable transformation T : C → C of Jordan boundary into itself, allowing to do some conclusions about properties of the Dirichlet problem in Ω. Let us describe it.
Let Ω be arbitrary bounded domain which is convex with respect to characteristics of the equation (1.1), i.e. it has the boundary C intersected in at most two points by each straight line that is parallel to xor y-axes. We start from arbitrary point M 1 on C and consider a vertical line passing through M 1 . Generally, there are two points of intersection with the curve C: M 1 and some point M 2 . We denote I 1 an involution which transform M 1 into M 2 . Then, starting from M 2 , we consider a horizontal line passing through M 2 . Let M 3 be the second point of intersection with the curve C. Let I 2 be corresponding involution: I 2 M 2 = M 3 . We then repeat this process, applying step-by-step involutions I 1 and I 2 . Denote T = I 2 I 1 , T −1 = I 1 I 2 . This transformation T : C → C gives us a discrete dynamical system on C, i.e. an action of group Z and each point M ∈ C generates an orbit {T n M |n ∈ Z}. This orbit can be finite or denumerable set. The point M with finite orbit is called a periodic point and smallest n, for which T n M = M , is called a period of the point M . In the paper [34] the uniqueness breakdown in the problem (1.1),(1.2) have studied in connection with topological properties of the mapping T for the case of even mapping T . The mapping T is called to be even or preserving an orientation if each positive oriented arc (P, Q) with points P, Q ∈ C transforms into positive oriented arc (T P, T Q). F.John have proved several usefull assertions, among of which we extract the following one.
Sufficient condition of uniqueness. The homogeneous Dirichlet problem (1.1), (1.3) in the bounded domain has only a trivial solution in the space C 2 (Ω) if the set of periodic points on C is finite or denumerable. It has been selected four possible cases of dynamical system behavior: I) all points are periodic (then their periods coincide); II) there are periodic and nonperiodic points; III) there are no periodic points and there is not a point, the orbit of which {..., T −1 P, P, T P, ..., T n P, ...} is dense in C; IV) there are not periodic points and there is a point, the orbit of which {..., T −1 P, P, T P, ..., T n P, ...} is dense in C (transitive case).
In the work [34] it has shown that for a C 2 -smooth curve the case III) is impossible. For the case II) it has proved that there is an arc D 0 on C such that each two arcs from D 0 , I 1 D 0 , T D 0 , I 1 T D 0 , T 2 D 0 , ..., I 1 T n−1 D 0 , T n D 0 , ... have not any common point. Note that for the analytical boundary this is impossible because in this case T is a diffeomorphism. In the case IV) it has proved that the Denjoy-Poincaré rotation ξ ( [46]) of the John mapping T is irrational and T is topologically equivalent to a turn of an unit circle about the angle πξ (i.e. there exists a homeomorphism h from C onto the unit circle S such that the mapping hT h −1 : S → S is a turn about the angle πξ).
For the case when every point of C is periodic it have proved the all periods are coincided. But here we don't know anything about solution uniqueness, although R.A. Alexandrjan have shown in this case [3] that there is a generalized solution of the problem (1.1),(1.2) which can be built as a piecewise constant function.
We will assume that the domain has boundary C satisfying the condition: The curve C is smooth and each characteristic line either doesn't intersect the curve C or touchs it at a point, named a vertex, or cuts C at two points. (2.1) Note that above John's condition of convexity with respect of characteristic directions will be fulfilled under the condition (2.1) on C = ∂Ω in the case of a bounded domain Ω.
For the case of bounded domain with a biquadratic boundary we will see that above sufficient uniqueness condition is also necessary, moreover, it will be so even for cases when the curve C is unbounded but then we should change the setting of the problem. Namely, along with the usual setting of the uniqueness property: the examined bounded domain is such that the homogeneous Dirichlet problem (1.1),(1.3) has only trivial solution in the space C 2 (Ω) ∩ C(Ω), (2.2) for cases when the curve C is unbounded we will examine the following modification of uniqueness property for the homogeneous Dirichlet problem : the examined curve C is such that each analytic in real sense solution in R 2 of the equation ( Proof. Let a function u ∈ C 2 (Ω) ∩ C(Ω) be a nontrivial solution of the problem (1.1),(1.3) in a domain Ω with the condition (2.1). As it is well known, there exist two functions u 1 , u 2 of the class C 2 depending on one variable such that u(x, y) = u 1 (x)+u 2 (y) which we will write for any point P ∈ C as u(P ) = u 1 (P )+u 2 (P ). For the case of the property (2.3) consider a domain Ω ⊂ R 2 of points P = (x 0 , y 0 ) ∈ R 2 for which there exists a pair of different points of intersection C ∩ {x = x 0 } of the curve with corresponding vertical line and also there exists a pair of different points of intersection C ∩ {y = y 0 } of the curve with the horizontal characteristic line. Then for any point P ∈ Ω using definitions we easily obtain: u 1 (I 1 P ) = u 1 (P ), From what follows that equalities u 1 (P ) = u 1 (T n P ), u 2 (P ) = u 2 (T n P ) hold for each integer n. Continuity gives us that u 1 (P ) = u 1 (Q), u 2 (P ) = u 2 (Q) for any point Q from closure of the orbit of P . Because for the transitive acting the closing of the orbit of any point coincides with C then u 1 ≡ const and u 2 ≡ const and, therefore, u ≡ 0 in Ω. In the case of the property (2.3) the analyticity allows us to continue the zero solution onto the plane.
Above we have noted that for any analytic boundary only two cases are admissible: periodic (I) and transitive (IV by John). Therefore we give the following setting of the periodicity problem for John mapping : What curve from a given class of curves has the property: T he John mapping T : C → C has at least one periodic point.
Then as we have known already, the all points are periodic on our curve. Along with above settings we will consider also a setting with complex John mapping. IfC ∈ C 2 is a biquadratic complex curve (i.e. one-dimensional complex variety (1.4)) then it is fulfilled the property of type 2.1: Almost each "vertical" line x = x 0 intersects the curveC at two different points and almost each "horizontal" line, too. (2.5) LetC ∈ C 2 be an analytic curve with the same property. Then we can construct a John mapping T in the same way as in real case. And we can ask a similar question about the periodicity problem for the complex John mapping : What curve from a given class of curves has the property: T he comp lex John mapping T :C →C has at least one periodic point. (2.6) Corresponding complex setting of the problem (1.1),(1.3) we will use is the following: To find two meromorphic functions f (x), g(y) of one complex variable such that f (x) + g(y) = 0 as soon as (x, y) ∈C ⊂ C 2 . (2.7) Note that for the case of transitive action of the mapping T the proof of the proposition 1 one can easy transfer onto this complex case: Proposition 2. For a biquadratic complex curve (1.4) if the complex John mapping T transitive acts onC then the complex problem (2.7) has only trivial solution.
Below we will consider biquadratic curves C satisfying the property (2.1) and will give an explicit criterion distinguishing the cases of periodicity and transitivity and in the first case we will build an explicit nontrivial solution of the problem (1.1), (1.3) in sense of the settings 2.2 or 2.3 and as an intermediate the setting (2.7). In the case of transitivity each solution is proved to be zero.
2.3. Boundary value problems and a moment problem. In this subsection we are going to indicate an equivalence of properties of some boundary value problems, in particular of the Dirichlet and Neumann problems, and give a moment problem which is responsible for this properties. Thus we will obtain some problems that are equivalent to the problem (1.1), (1.3) in the setting (2.2). For details and generalizations see [16] or [17]. This equivalence is based on a connection condition of solution traces for the Cauchy problem.
And let's consider an overdetermined boundary value problem for the equation that is otherwise written down the Cauchy problem u| ∂Ω = γ 0 , u ′ ν | ∂Ω = κ 0 . We ask a question: what is a connection between the functions γ and κ if they are generated by a solution u of the equation (2.8)?
In order to answer we need the following construction.
The equation (2.8) we write down also as where a j = (a j 1 , a j 2 ), j = 1, 2 are unit real vectors. Let's enter vectorsã 1 = (−a 1 2 , a 1 1 ),ã 2 = (−a 2 2 , a 2 1 ), directing vectors of set of characteristic directions Λ = Λ 1 ∪ Λ 2 , Λ j = λã j |λ ∈ R , j = 1, 2, ã j , a j = 0. Let ϕ 0 = ϕ 1 − ϕ 2 where ϕ j is any solution of equation tan ϕ j = λ j , i.e. ϕ j is an inclination angle of vector of the characteristic direction corresponding to the root λ j , ϕ 0 is angle between characteristics, and let ∆ = sin ϕ 0 = det a 1 a 2 , a j are columns. Here and below a vector of a characteristic direction is understood as a vector ν ∈ C 2 which is a null of the symbol: l(ν) = 0. The traces γ and κ of a solution u are linked by the following relations.
where x(s) is a moving point on ∂Ω.
Consider following a moment problem: where on two given vectorsã j ∈ R 2 and on two sequences of numbers µ j N it is found the function α. Obviously, for the case when ∂Ω is the unit circle and vectorsã j , j = 1, 2 are equalã 1 = (1, i);ã 2 = (1, −i) this moment problem turn on well-known trigonometric moment problem because then (x(s) ·ã j ) N = exp(±iN ). Another way to the same is to write the Chebyshev polynomial T N insteed of the power.
Among a lot of problems connected with above moment problem we will consider the problem of indeterminacy (uniqueness): for what curve ∂Ω and vectorsã j , j = 1, 2 there exists a function α such that (2.14) The following fact takes place  Proof. 1) ⇒ 2). Using pair γ = 0, κ = 2α/∆ with the help of the statement 1 we build the solution u ∈ H m−q (∂Ω).
Note that instead of the Neumann problem in the statement 3 we can write the boundary condition of view u ν * = λu γ (2.15) with an arbitrary constant λ.
Note also that the case when the domain Ω is an ellipse has examined in the works by A. Huber [32], R. Alexandrjan [4], V.I. Arnold [6] and one of the authors [16] or [17]. An answer to the question about properties of such the Dirichlet problem   (2.15), in particular of the Neumann problem, and is equivalent to the existence of a nontrivial solution of the moment problem (2.14). Below we will give a criterion of nontrivial solvability each of these problems with the curve (1.4) in a view of the condition (2.16).

2.4.
Duality equation-domain and one more equivalent problem. Let Ω ⊂ R n be a bounded semialgebraic domain given by means of the inequality Ω = {x ∈ R n : P (x) > 0} with a real polynomial P . Equation P (x) = 0 gives us the boundary ∂Ω. We assume the boundary of domain Ω is nondegenerate: | ∇P | = 0 on ∂Ω. Consider the Dirichlet boundary value problem for the equation (2.8) of the order 2 with constant complex coefficients: where D x = −i∂/∂x. We understand a duality equation-domain as a correspondence the problem (2.17) and the equation: given in the following statement: The class Z is determined here as space of Fourier images of functions of the form θ Ω v, where v ∈ C 2 (R n ) and θ Ω is the characteristic function of the domain Ω.
For a clearness we give here a sketch of the proof. We assume the problem has a nontrivial solution u in C 2 (Ω), letũ ∈ C 2 (R n ) be any smooth continuation of u on the R n then we multipleũ on characteristic function θ Ω of domain Ω (θ Ω = 1 in domain Ω and θ Ω = 0 out Ω) and apply the operator L to the product θ Ωũ . Differentiating the product by Leibniz we obtain where δ ∂Ω is measure supported on ∂Ω: < δ ∂Ω , φ >= ∂Ω φ(x)ds x , L 2 (u) = L(ν)u, ν is the external normal as earlier. The first term in (2.19) is equal to zero by means of equation. Take into account the boundary condition u ∂Ω = 0, obtain the last term will be transformed into term of view of the second term: L 1 (u, ∇u) δ ∂Ω as, for example, for the case of one variable xδ ′ (x) = −δ. Then we multiple the obtained equality on P (x), the right-side part vanishes because e.g. for the case of one variable xδ(x) = 0. We obtain an equation P (x)L(D)(θ Ωũ ) = 0. Applying the Fourier transform F we obtain the equation (2.18) where w = F (θ Ωũ ). Necessity is proved. Sufficiency will be obtained by means of conversion of this proof. A full proof for a general case see in the works [17] or [15].
The term 'duality equation-domain' means here an equivalence of the problems (2.17) and (2.18) that we will read here as: Now from statement 5 it follows and it is conversely.
The last boundary value problem for the equation of the fourth order has only two boundary condition insteed of fourth condition as it should be e.g. for the problem of a Goursat (or Darboux) type. Therefore it is no wonder that a nontrivial solution of the problem (2.22) exists there. But as we will show below for almost each curve (1.4) the problem (1.1), (1.2) has only trivial solution, therefore almost each problem (2.22) has only trivial solution as well. But this statement can seem astonishing namely because of the insufficiency of the data, in spite of the stipulation that the solution v belongs to the space Z.
We finished the statement of propositions on boundary value problems for general domains and now we should wait for section 6 in order to obtain explicit answers for the Dirichlet problem in domains with biquadratic boundary.

Generic biquadratic curve
3.1. Parameterizations of generic biquadratic curve. The complex curve (1.4) is remarkable by that this is the most general algebraic curve with the property that almost each vertical or horizontal line intersecting C intersects it in 2 points.
Let (1.4) be a generic nondegenerate real biquadratic curve. Assume that the parameters a ik are chosen such that the real curve C bounding the domain satisfies the condition (2.1) of the subsection 2.2.
We will begin our study of the problem (1.1),(1.3) by an observation that the curve (1.4) is elliptic curve allowing an uniformization in terms of elliptic functions.
Indeed, rewrite the equality (1.4) in one of the form We multiply the equation (3.1) by A 2 , the equation (3.2) by B 2 and reduce these expressions to the forms are birational transformations and D 1 (x), D 2 (y) are the discriminants of quadratic equations (3.1) and (3.2): . In a general situation the discriminants D 1,2 are polynomials of 4-th or 3-d degree.
Recall [60], that every curve of the kind with a generic 4-th degree polynomial π 4 (x), can be transformed to a canonical form in terms of the elliptic Weierstrass function ℘(t) with primitive periods 2ω 1 , 2ω 2 . The parameters g 2 , g 3 are so-called (relative) invariants of the polynomial D 1 . They are real for real π 4 .
As (x, Y ) → (X, y) is a birational transformation, the primitive periods of the curves (3.3) are coincide. The invariants g 2 , g 3 may be found by periods. Hence we obtain the following important statement (which was mentioned by Halphen [31]): Statement 6. Invariants g 2 , g 3 of polynomials D 1 (x) and D 2 (x) are the same.
Thus, both curves (3.3) are elliptic ones [60] with appropriate primitive periods 2ω 1 , 2ω 2 , the curve (1.4) is homeomorphic to the torus: C ≈ C/(2ω 1 Z ⊕ 2ω 2 Z) and there are standard structures of a commutative group and an Abelian variety [29]. We deal with elliptic functions of the second order. Recall [60] that properties of a general elliptic function can be characterized by the number of poles (or that is equivalently, zeroes) in the fundamental parallelogram of periods. This number is called an order of the elliptic function (the order is taken with account of multiplicity of poles). By the Liouville theorem, the simplest possible order is 2 [60]. E.g. the Weierstrass function ℘(t) is of degree 2 because it has the only a pole of multiplicity 2 at the point t = 0 of the fundamental parallelogram.
The general elliptic function of the second order Φ(t) has two arbitrary poles p 1 , p 2 and two arbitrary zeroes ζ 1 , ζ 2 in the parallelogram of periods. The only condition is [60] where Ω = 2m 1 ω 1 + 2m 2 ω 2 is an arbitrary period. It can be easily showed that generic elliptic function of the second order with given periods 2ω 1 , 2ω 2 can be presented as Thus, Φ(t) depends on 4 independent parameters, say α, β, δ, t 0 . There is another, sometimes more convenient, representation of the function Φ(t): where κ is a constant and parameters e 1 , e 2 , d 1 , d 2 are related as The form (3.8) is obtained from standard representation of the arbitrary elliptic function in terms of the Weierstrass sigma-functions [60]. In this case the points e 1 , e 2 and d 1 , d 2 coincide with zeros and poles of the function Φ(t) and the condition (3.9) is equivalent to a balance condition between poles and zeros of the generic elliptic function. Note that apart from ℘(t) there are another special cases of functions of second order, e.g. the Jacobi elliptic functions sn(t; k), cn(t; k), dn(t; k) [60] that we will use below.
What is an uniformization for the biquadratic curve? The answer is given by following Theorem 1. The generic complex biquadratic curve (1.4) can be parameterized by a pair of elliptic functions of the second order with the same periods: Conversely, any two elliptic functions x = Φ 1 (t), y = Φ 2 (t) of the second order with the same periods satisfy an equation (1.4).
This theorem is proved essentially e.g. in the famous Halphen monograph [31] on elliptic functions. We give a proof based on some main Halphen's ideas here.
As we already saw, the polynomials D 1 (x) and D 2 (y) have the same invariants g 2 , g 3 . Hence they both can be reduced to the same canonical Weierstrass form (see, e.g. [1], s.34) by means of a pair of the Möbius transforms with some complex parameters µ 1 , . . . η 2 . Hence the equation (3.11) becomes because dỹ/dx = −X/Ỹ as above. But the equation (3.13) means that for appropriate periods 2ω 1 , 2ω 1x where ℘(u) = ℘(u; ω 1 , ω 2 ), u is an uniformization parameter and u 0 is a complex constant. Now we can return to initial variables x, y by means of inverse Möbius transforms to get finally Then we apply formula (3.7) which says that we indeed obtained a pair of elliptic functions of the second order.
The inverse statement of the theorem follows from a general theorem that any two elliptic functions x(t) and y(t) with the same periods satisfy an algebraic equation F (x(t), y(t)) = 0. The degrees of the polynomial F (x, y) with respect to variables x and y are determined by orders of corresponding elliptic functions. If both functions have the order 2 then the polynomial F (x, y) has the degree at most 2 with respect to each variable (this statement is contained in [60] as a problem for the reader).
Thus, we proved the theorem 1. Moreover, simultaneously we have proved the following Proposition 4. There exist complex linear-fractional transforms (3.12) such that transformed generic curve (1.4) having the same form can be parameterized by only the Weierstrass function ℘(u) as in (3.14).
Considering the formulae (3.10) we can write down our parameterization in terms of the Weierstrass sigma-function: with two restrictions e 1 + e 2 = d 1 + d 2 and e 3 + e 4 = d 3 + d 4 . If now we make a shift t → t − t 0 of the uniforming parameter t then we can choose t 0 such that, say,

This means
Proposition 5. In formulae (3.10) the function Φ 1 (t) can be chosen even: From our considerations it follows an important corollary (which was mentioned by Halphen [31] as well): Proposition 6. Consider the differential equation (3.11). Let D 1 (x) and D 2 (y) be polynomials of degree 4 or 3 with the same invariants g 2 , g 3 . And let (x(t), y(t)) be a solution of this equation (parameterized e.g. by an initial condition). Then x(t), y(t) satisfy a biquadratic equation of the form (1.4).

Biquadratic foliation and singular points.
There is an interesting mechanical interpretation of last results. Assume that we have a dynamical Hamiltonian system for two canonical variables x(t), y(t) satisfying a system of equationṡ where H(x, y) is a Hamilton function of the system. Obviously H(x, y) is an integral of the system (3.17), i.e. (H(x, y)) · = 0. Choose the Hamiltonian as the biquadratic function (1.4): H(x, y) = F (x, y). We have F (x, y) = c with some constant c depending on initial conditions for x, y. This constant can be incorporated to the coefficient a 00 , so we can write downF (x, y) = 0, whereF (x, y) = F (x, y)−c is again a biquadratic curve (note that forF (x, y) the coefficients remain the same whereas the coefficientsÃ 0 (y) andB 0 (y) differ from initial ones by a constant). Then We thus see that for any fixed Hamiltonian level H = c the variables x(t), y(t) satisfy the Euler equation (3.11) where polynomialsD 1 (x),D 2 (y) have the same invariants g 2 ,g 3 . Note, that in this case the invariantsg 2 ,g 3 (and hence the periods 2ω 1 , 2ω 2 ) will depend on the value of integral c. We thus obtain a whole one-parameter family of biquadratic curves F (x, y) = c and corresponding elliptic functions x(t), y(t) of the second order that are trajectories of this dynamical system. Rewrite discriminantsD 1 (x),D 2 (y) in factorized forms where q 1 , q 2 are leading coefficients of the discriminants and x i , y i , i = 1, 2, 3, 4 are their roots (for simplicity we assume that both discriminants have degree 4). In general, roots x i , y i will depend on the parameter c. What is mechanical meaning of points x i , y i ? From equations of motion it is seen thaṫ Thus x i and y i are stable points:ẋ i =ẏ i = 0. We should demand that the points (x i , y k ) belong to our biquadratic curveF (x, y) = 0. This leads to conditions that points (x i , y k ) satisfy the conditions which in turn means that points (x i , y k ) are singular points of the biquadratic curvẽ F (x, y) = 0. As we saw, in generic situation this curve is elliptic and hence has a genus 1 (genus C = C 2 n−1 − d, n is degree and d is a number of double points of C). The degree of this curve is 4. Assume that the curve is irreducible. Then such curve cannot have more than 3 singular points in complex projective plane. The latter is defined by the coordinates (s 0 , s 1 , s 2 ) such that x = s 1 /s 0 , y = s 2 /s 0 . In these coordinates we have equation of our curve Elementary considerations show that two points (0, 1, 0) and (0, 0, 1) of the projective plane are singular for any values of parameters a ik . Thus, only one singular point can exist in each finite part of the plane. In turn, this means that at least two roots, say x 1 , x 2 of the discriminantD 1 (x) should coincide: x 1 = x 2 . The same condition holds for the discriminantD 2 (y), i.e. y 1 = y 2 because invariants g 2 , g 3 of the discriminantsD 1 (x) andD 2 (y) are the same. Then it is elementary verified that the point (x 1 , y 1 ) will be indeed a singular point of the biquadratic curve. In principle, the second singular point can occur. But in this case the genus of the curve will be -1 meaning that the curve is reducible.
We thus have the following Proposition 7. The irreducible biquadratic curveF (x, y) = 0 has the a singular point in a finite part of the (complex) plane if and only if the discriminantD 1 (x) (and henceD 2 (y) as well) has a multiple zero x 1 (correspondingly y 1 ). In this case the point (x 1 , y 1 ) is singular and unique.
Note this proposition can be reformulated in an equivalent form. Indeed, the polynomial D 1 (x) has a multiple zero if and only if its discriminant is zero. Thus, in order to find all singular points of the curve we should first calculate the discriminant D 1 (x) of the biquadratic curve F (x, y) = 0 and then calculate the discriminant ∆(D 1 (x)) from the polynomial D 1 (x) ("discriminant from the discriminant"). We have obviously coincidence of the two discriminants ∆(D 1 (x)) = ∆(D 2 (y)) because the invariants g 2 , g 3 of the polynomials D 1 (x) and D 2 (y) are the same (statement 6). Condition ∆(D 1 (x)) = 0 (or, equivalently, ∆(D 2 (y)) = 0) gives us some nonlinear equations for the coefficients a ik . Under such condition the biquadratic curve F (x, y) = 0 has a genus < 1, i.e. it is either irreducible and has the only singular point in a finite part of the projective plane, or it is reducible: are two polynomials linear with respect to each argument x, y (but τ k (x, y) are not in general linear functions, they may contain the terms like xy). We have obtained Proposition 8. The condition ∆(D 1 (x)) (= ∆(D 2 (y))) = 0 is necessary and sufficient for the equality: genus C = 1.

3.3.
Case of generic symmetric curve. Above we considered generic case when our biquadratic curve F (x, y) = 0 is non-symmetric.
Assume now that our curve is symmetric, i.e. F (x, y) = F (y, x). Equivalently, this means the coefficient matrix a ik in (1.4) is symmetric: a ik = a ki . In this case, obviously, both discriminants coincide D 1 (x) = D 2 (x). From the Euler differential equation (3.11) we obtain that a parameterization can be given by formulae where u 0 is a constant and Φ(u) is an even function of the second order, i.e. Φ(t) can be presented in the form Thus, in the symmetric case a parameterization is provided by some even elliptic function of the second order. Vice versa, any pair (x, y) = (Φ(u), Φ(u + u 0 )) with arbitrary u 0 generates a symmetric biquadratic curve by formulas (3.20) because the point (y, Note that the last statement was attributed to Euler in the work [59]. We obtain the following If the initial curve is real then it can be transformed to real symmetrical curve also (although corresponding transformation can be with complex coefficients).
Proof. As we have shown in the theorem 1 (see proposition 4), generic non-symmetric curve can be transformed to a curve described as (3.14) by linear-fractional complex changes (3.14) that means the curve-image is symmetric one in virtue of the last proposition. In the real case we note that as it is well-known, the invariants g 2 , g 3 of real polynomial D 1 (x) are real, so that the differential equation (3.13) is real and there exists its real solution (x(u), y(u)) that can be extended onto complex domain and therefore satisfies the symmetric equation (1.4). The last equation has irreducible polynomial which must be real if a 22 = 1 because we always can choise real parameters u 1 , ...u 8 such that vectors V i composed of components x k (u i )y l (u i ), 0 ≤ k, l ≤ 2, 0 ≤ k + l < 4 will be linear independent, then the coefficients of our polynomial satisfy a linear system of 8 linear equations with real coefficients (V i ) j and the real right-side parts −x 2 (u i )y 2 (u i ).
Note that another proof of the last fact is there in the work [35], see below statement 7.
Let us now consider the question how restore the polynomial F if we know the discriminants D 1 , D 2 . In symmetric case T. Stieltjes [55] proposed a nice explicit formula for the polynomials F (x, y) by means of a solution of the differential equation (3.11). Assume that Then F (x, y) can be given by the determinant: The function F (x, y) is defined up to an arbitrary non zero number factor. It is assumed that the curve is nondegenerated, i.e. its genus is 1. This is possible if and only if the determinant is nonzero. C is an integration (arbitrary) constant ( [55]). The Stieltjes formula is useful in the problem of reducing of the arbitrary symmetric biquadratic curve to some standard forms, as we will see below.
3.4. Canonical forms of biquadratic curve. The general curve (1.4) contains 8 free parameters. It is naturally to transform this curve to the form containing the smallest possible numbers of parameters. First, note that under arbitrary projective transformations with complex parameters we obtain similar equation (1.4) but with transformed parameters a ik . This idea has already exploited in the proof the theorem 2. Every projective transformation (3.23) contains 3 independent parameters, hence it is possible to reduce the total number of independent parameters a ik to 8 − 6 = 2. As our curve is an elliptic one these free parameters are only invariants g 2 , g 3 under linear-fractional transformations of variables x and y separately. More explicitly, if we consider the projectivisation 2 i1,i2,j1,j2=1 a i1i2j1j2 x i1 1 x i2 2 y j1 1 y j2 2 , x = x 1 , y = y 1 of initial curve (1.4) and its projective transformations of variables x and y separately that g 2 , g 3 are only invariants (of respective wights 4 and 6) but for the transformations of the group SL(2, C) these are absolute invariants.
Above we reduce the general case to the symmetric one so that we restrict ourselves with considering a symmetric curve F (x, y) = F (y, x). So we would like to find some canonical forms of the curve containing only 2 parameters.
There are two obvious canonical forms which can be obtained for the curve F (x, y) = 0. These two forms correspond to two canonical forms of elliptic integrals in the Euler differential equation (3.11).
(I) The first one can be obtained if one reduces polynomial D 1 (x) = D 2 (x) to the canonical cubic Weierstrass form: Such form can be always achieved by an appropriate projective transformation (with possible complex coefficients). Then from the Stieltjes formula (3.22) we obtain the expression (see also [31]) where w = C is an arbitrary parameter. There is a parameterization of this curve in terms of the Weierstrass elliptic function and w = ℘(u 0 ), where u 0 is an arbitrary constant. It is easy to calculate the discriminant: is a polynomial of degree 1 with respect to each variable x, y. If 4w 3 − g 2 w − g 3 = 0 then the curve is irreducible. The singular points in a finite part of the complex projective plane appear only if g 3 2 = 27g 2 3 . This condition means that the polynomial 4x has a double root, say e 1 = e 2 . In this case we can put g 2 = 3τ 2 , g 3 = τ 3 , where τ is some parameter. Then it is elementary verified that the point x = y = −τ /2 will be the only finite singular point of the curve F (x, y) = 0. If g 3 2 − 27g 2 3 = 0 then there are no singular points in the finite part of the projective plane and the curve (3.24) is irreducible and has a genus 1.
The curve (3.24) contains 3 parameters w, g 2 , g 3 . Assume that 4w 3 − g 2 w − g 3 = W = 0. In this case the curve is irreducible. By the linear transformation of arguments x → αx + w, y → αy + w, where α 3 = 1/W we can eliminate terms x 2 y, xy 2 and x 2 + y 2 . The curve then is reduced to the form containing only 2 independent parameters A, B. The curve is elliptic non-singular (i.e. having genus 1) if condition holds. Thus we proved the following Proposition 11. The generic complex biquadratic curve (1.4) can be transformed to canonical form (3.25) by means of a linear-fractional complex changes of variables (3.12). If the initial curve is real then it can be transformed to real form also (although corresponding transformation can be with complex coefficients).
As we saw, any irreducible biquadratic curve can be transformed to this form. However, here we need general projective transformations (3.23) with complex parameters. This will be so even in the case when all parameters a ik of the biquadratic curve are real.
(II) In beginning let remember well-known Legendre transformation. Assume that an elliptic curve has already the form Y 2 = π 4 (x) (see (3.3)), where π 4 (x) is an arbitrary polynomial of the 4-th degree with real coefficients. Using only a linear-fractional transformation x = Γ( x) with real coefficients it is possible to reduce this curve to the form (with a new Y ) where π 4 (x) has only even degrees of x: π 4 (x) = αx 4 + βx 2 + γ with some real coefficients α, β, γ (see, e.g. [23]).
Assume now that our biquadratic curve is symmetric and has only real coefficients a ik . Then we can always transform this curve to a form without terms of odd degrees, i.e. a 21 = a 12 = a 10 = a 01 = 0. Indeed, taking x = y in expression (1.4) for F we obtain the polynomial F (x, x) of fourth degree and apply to it the Legendre change x = Γ( x) in a generic case that reduces the equation F (x, x) = 0 to the view α x 4 + β x 2 + γ = 0. Therefore the change with the same Γ x = Γ( x), y = Γ( y) (3.27) applied to (1.4) gives us an equation with desired polynomial (observation from [35]).
We then obtain so called the Euler-Baxter biquadratic [10], [58]: where a, b, c are remaining real parameters if the initial curve was real and, obvious, will be complex if the initial curve was complex. The curve (3.28) plays the crucial role in deriving addition theorem for the elliptic function sn(t) [1]. It appeared also in Baxter's approach to 8-vertex model in statistical mechanics [10]. We first analyze possible finite singular points of the curve (3.28). The discriminant of the curve (3.28) is equal to If c = 0 then we have an obvious singular point x = y = 0. This singular point will be isolated for all values of the parameters a, b excepting the case a ± b = 0. If a ± b = 0 then the curve F = 0 becomes reducible. In any case if c = 0 the genus of the curve (3.28) is less then 1 (i.e. 0 for generic case and −1 for exceptional case corresponding to a reducible curve). Thus in this case the curve is not elliptic and can parameterized by rational functions. Here the change x → κ/x, y → κ/y gives us a form F = x 2 + y 2 + axy ± 1. Using then scaling transformations x → κx, y → κy in real case we can reduce the coefficients c = 0 to ±1 depending on sign of this coefficient.
Proposition 12. The generic complex biquadratic curve (1.4) can be transformed to canonical form (i) by means of a linear-fractional complex changes of variables (3.12). If the initial curve is real then it can be transformed to real form (i) or (ii) (although corresponding transformation can be with complex coefficients). By means of only some real linear-fractional transformation (3.12) any generic real symmetric biquadratic curve (1.4) can be reduced to one of two form: (i), (ii).
Remark that in the paper [35] it is proved the following Statement 7. By means of only some real birational transformation any generic real elliptic biquadratic curve (1.4) can be reduced to one of three form: (i), (ii) or to the form (iii) F = x 2 y 2 + a(x 2 − y 2 ) + 2bxy − 1.
Note that the discriminants of the curve (iii) are equal to D 1 (x) = −a(x 4 −bx 2 + 1); D 2 (y) = a(y 4 +by 2 + 1);b = b 2 + a 2 + 1 a . (3.30) In the complex case we can reduce the coefficients c = 0 to 1 and leave only the case (i) among nondegenerate cases. Here following [10] we can find a parameterization of the curve (3.28) in terms of elliptic Jacobi function sn: Proposition 13. The curve (3.28) can be parameterized by the formulae with any sigh ±, where parameters k, η are determined by relations Note that we can replace t =t + K in order to deal with an even function. Note that the mapping (3.31) is an analytic diffeomorphism from fundamental parallelogram, factored by a standard way into torus, ontoC.
Real parameterizations of curves (i), (ii) and (iii) are brought in [35] and [36]. Now let's remember expressions (3.3) writing as We deduce y-extreme point (x 1 , y 1 ) gives us 0 = y ′ ( . Thus, for any y-vertex (x 1 , y 1 ) the number y 1 is a root of the polynomial D 2 (y 1 ). Conversely, from the equalities (3.33) and y ′ F y = −F x we see that for each such root y 1 of D 2 the point (x 1 , y 1 ) is either y-vertex (extreme point in y-direction) or a singular point. But as we have seen in subsection 3.2 a singular point can be only if a root of D 2 is multiple, i.e. ∆ = g 3 2 − 27g 2 3 = 0; this is an exceptional case. Therefore it is proved the following Proposition 14. For generic case the roots of D 2 (y 1 ) and only they are y-coordinates of y-vertexes of the curve. It's analogous, for generic case the roots of D 1 (x 1 ) and only they are x-coordinates of x-vertexes. Now we can give geometrical an interpretation of cases (i), (ii), (iii), a proof of which can be easy obtained by analysis of formulae (3.29) and (3.30).
3). In the case (ii) the equation D 1 (x) = 0 (and the equation D 2 (y) = 0) has two real roots (there are 2 x-vertices and 2 y-vertices). 4). In the case (iii) for a > 0 (b > 2) the equation D 1 (x) = 0 has 4 real roots and the equation D 2 (y) = 0 has no real root (there are 4 x-vertices and no y-vertex). 5). In the case (iii) for a < 0 (b < −2) the equation D 1 (x) = 0 has no real root and the equation D 2 (y) = 0 has four real roots (there are 4 y-vertices and no x-vertex).
If we will allow values x = ∞, y = ∞ i.e. consider our real curve on a thorus S 1 × S 1 then the number of x-vertexes and the number of y-vertexes will not be changed by Möbius transformations (3.23), therefore we obtain Proposition 16. Linear-fractional changes (3.23) don't remove the curve from its class 0)-5) of the proposition 15.
3.5. John mapping of biquadratic curve and periodicity. At first, we show how complex John algorithm works for the curve C. Consider a case of some more general curve.
LetC ⊂ C 2 be a complex curve described parametrically as where ε is a nonzero complex parameter. Assume that φ(z) is an even periodic meromorphic function, i.e. φ(−z) = φ(z), φ(z + T ) = φ(z) with some (complex) constant T . And let C =C ∩ R 2 be a real curve given by means of a contraction of the functions x, y on some set S ∈ C. We will assume that the curveC satisfies the condition: The curveC is nondegenerate and each straight line of the form x = x 0 or y = y 0 intersects the curveC at no more than two points (3.35) Replace t → −t. Then, due to the fact that φ(t) is even we see that x(−t) = x(t) but in general y(−t) = y(t). This means that the point (x(t), y(−t)) on the curve is obtained as the second intersection point of C with the "vertical" line passed through the initial point (x, y). Thus the transformation t → −t is equivalent to the involution I 1 in the John algorithm. Quite analogously, the transformation t → −t − 2ε remains the coordinate y on the curve C unchanged whereas x is transformed to another point on the intersection of the curve C with "horizontal" line. Thus, transformation t → −t − 2ε is equivalent to second involution I 2 in the John algorithm. For the John mapping T we have obviously with some integer n, m.
In the case of the biquadratic curve we have two periods 2ω 1 , 2ω 2 and then the periodicity condition is taken the form: with some integer n, m 1 , m 2 . Now we have only two cases: either all points of C are not periodic (for the John mapping) or each point is periodic with the same period (that is equal to n if the condition (6) or (3.38) is fulfilled).
Consider the last canonical form of previous subsection -the Euler-Baxter curve (3.28) that is parameterized as (3.31). It is important that for given x = √ k sn(t; k) there are two values of y corresponding to two values of η: y 1 = √ k sn(t + η; k) and y 2 = √ k sn(t − η; k). These two values y 1,2 correspond to two points of intersection of the line x = x 0 with the curve (3.28).
Thus we can find all points M n of the complex John algorithm: The periodicity condition with a period n is with some integer m 1 , m 2 and constants K = K(k) In order to write analogous formulae in real case we should write real paramerizations by means of real-valued elliptic functions.
Consider first the case a > 0, c = +1 of real x-y−symmetric curve (3.28) and use the parameterization (3.31). Above, in the proposition 15, (s. 0) we noted that under a > 0 the condition b 2 > (a+1) 2 (orb > 2) is necessary and sufficient for the existence of real curve (3.31). Therefore assume b 2 > (a + 1) 2 then k + k −1 ≥ 2 and we may choose k = k 1 , 0 < k 1 < 1. From the second equality (3.32) we obtain that sn(η, k) is pure imaginary. This means η = 2mK + θi with an integer m and real θ (see [9]). Moreover, x must be real, that is either t = nK ′ i + τ or t = (2n + 1)K + iτ with an integer n and real τ ( [9]). But y must also be real, i.e. t + η = K ′ n 1 i + τ 1 or t + η = (2n 1 + 1)K + iτ 1 with an integer n 1 and real τ 1 . Let us add expressions for t and η, obtain that it is possible only a variant with the second formulae for t and t + η: τ 1 = θ + τ . Thus we have suppose that the parameter t isn't real: t = ±K + iτ , however the values x, y in the paramentization (3.31) are real and the parameter τ is real, here the sign before K determines a branch of the curve which is two bounded ovals in this case. The relations (3.32) have a denumerable set of solutions η, we choose θ = Im η as a minimal positive number of all Im η. Now the periodicity condition (3.40) can be written in the form (m = m 2 ): where from relations (3.32) we obtain that the number θ is a minimal positive solution of the equation sc(θ, k ′ ) = 1/ √ ak (here and below sc = sn/cn, ns = 1/sn and so on as usually). We proved the foolowing Consider also another cases of real x-y−symmetric curves (3.28). We take advantage of a list of cases given in the work [35] (note that the last work constains mistakes in above case c = 1, a > 0, here we used formulae from [10]). We write final parameterization formulae where the modulus k is given byb (k ′2 + k 2 = 1)) and the shift η − by a or b (they are coordinated). Hereb = (b 2 − a 2 − c)/a.

The Poncelet problem
In this section we demonstrate a nice correspondence between the mapping by F.John and famous the Poncelet problem (the Poncelet porism) for two conics. We start by recalling the Poncelet porism in a well-known form.

The Poncelet porism in form of two circles. At the beginning let a circle
A lie inside another circle B. From any point on B, draw a tangent to A and extend it to B. From the intersection point, draw another tangent, etc. For n tangents, the result is called an n-sided Poncelet transverse. This Poncelet transverse can be closed for one point of origin, i.e. there exists one circuminscribed (simultaneously inscribed in the outer and circumscribed on the inner) n-gon. We could begin with a polygon that is understood as the union of a set of straight lines sequentially jointing a given cyclic sequence of points (vertices) on the plane. If there exist two circles, inscribed and circumscribed for this polygon, then this polygon is called a bicentric polygon. Note that sides of the polygon can intersect and the intersection point is not obligatory to be a vertex. Furthermore, the inscribed circle does not obligatory touch a segment between vertices, the contact point can lie on extension of the side and therefore the circles can intersect. Bicentric polygons are popular objects of investigations in geometry. This is most known form of the Poncelet porism. If we denote by r the radius of the inscribed circle, by R the radius of the circumscribed circle and by d a distance between the circumcenter and incenter for a bicentric polygon then these three numbers can not be arbitrary and together with n they satisfy some relations. So, for the case of triangle the relation is sometimes known as the Euler triangle formula R 2 − 2Rr − d 2 = 0. One of popular notations for such relations (necessary and sufficient for existence of a bicentric polygon) is given in terms of additional quantities So, for a triangle above the Euler formula has the view: a + b = c, for a bicentric quadrilateral, the radii and distance are connected by the equation The relationship for a bicentric pentagon is 4(a 3 + b 3 + c 3 ) = (a + b + c) 3 . In a general case one introduses numbers and then the relationship can be written by means of elliptic functions in the form

Setting of the Poncelet problem.
Recall the Poncelet problem [13] for the case of two ellipses, for simplicity and as it was introduced by Jean-Victor Poncelet himself. We take two arbitrary ellipses A and B, A inside B. Let us have an arbitrary point Q 1 on the ellipse A and pass a tangent straight line to A at the point Q 1 . This tangent crosses the ellipse B at two points P 1 and P 2 , P 1 before P 2 with respect to a standard orientation. Then we take the point P 2 on B and pass the second tangent to the ellipse A. We denote as Q 2 the point on A where this tangent contacts with A. This tangent meets the ellipse B in two points P 2 and P 3 . Take the point P 3 and repeat this procedure. Then we obtain a mapping U B : B → B which acts by the rule U B : P k → P k+1 that will be called the Poncelet mappings below. Moreover, we obtain the mapping U A : A → A acting by the rule U A : Q k → Q k+1 . More precisely, because the definition of U B exploits a point Q ∈ A we should introduce two mappings I A , I B :C →C, C := {(Q, P ) ∈ A × B | P lies on tangent line to A at Q}, acting by the rules I A : (Q 1 , P 2 ) → (Q 2 , P 2 ), I B : (Q 1 , P 1 ) → (Q 1 , P 2 ). These mappings I B , I A generate a composition U = I B • I A that is similar to the John mapping T . We obtain also two sets of points P n and Q n on the ellipses B and A, respectively. The mapping U B has an inverse and generates a discrete dynamical system or, in other words, an action of the group Z on B as above for the John mapping. An orbit of this action is the set of the points P k , k = ..., −1, 0, 1, 2, ... and P k = U k−1 B P 1 . Now note that in a general case of a disposition the ellipses can be intersecting, then we must start from a point Q 1 on the ellipse A which is inside B and can determine the mappings U B , U A in the same way. In this case we can encounter with that the tangent straight line intersects the ellipse B only at one point, then we must consider this point as double. The ellipses can be tangent in one or two points, they can be tangent and intersecting simultaneously, they can be nonintersecting and lie one outside other, finally, they can be arbitrary irreducible conics. The way is the same. Note that one can consider the case the conic B is reducible, e.g., it is two nontangential different straight lines. Note more, that each projective transformation of the plane transforms a Poncelet mapping of conics into the same mapping of their immages, therefore we could restrict ourself to a case when one of conics is the unit circle.
The first interesting problem is to describe these crossing points explicitly. It was solved by Jacobi and Chasles who showed that the sequences P n and Q n can be parameterized by the elliptic functions. The second problem, the so-called Poncelet porism or the big Poncelet theorem (see [13]), consists in showing that if a particular trajectory of action on conics is closed (i.e., if P N = P 0 for some N > 2) then this property does not depend on the choice of the initial point Q 1 on the conic A (or on the choice of P 0 ). A modern treatment of this problem from an algebro-geometric point of view can be found, e.g., in [28]. Our approach is another. If we introduce the standard rational parameterizations of our conics (see below (4.3), (4.4)) then the parameters x and y of the points Q 1 , P 1 prove to be connected by a polynomial equation F (x, y) = 0. In a generic situation F (x, y) is a polynomial of the exact degree 2 with respect to each variable. Indeed, for a nondegenerate situation a tangent to the conic A at a point x should intersect B in two distinct points, and, moreover, from a point y on B there are two distinct tangents to A. Conversely, for any polynomial F (x, y) of the degree 2 in x and y it is possible to find two conics A and B parameterized as in (4.3) and (4.4). These arguments are sufficient to build trajectories of the John dynamical system. Note that similar considerations were exploited in [25] in order to show that the tangent and intersection points belong to an elliptic curve. The authors of [25] also introduced two involutions I 1 and I 2 which are equivalent to our ones in the John T -algorithm for the Euler curve. In our approach we derive the curve F (x, y) = 0 explicitly and study it. Note connections with Gelfand's question [38] and elastic billiard [58], [41].

4.3.
Passage to the John mapping on a biquadratic curve. Find an explicit expression for the Poncelet mapping U B . We introduce the standard rational parameterization of an arbitrary conic [13] that can be found even if the conic is reducible. Assume that the conic A is described by the coordinates ξ 0 , ξ 1 , ξ 2 of a two-dimensional projective plane, that is and, as it is well-known, the conic is irreducible iff the matrix A is nondegenerate. Note that we will describe vectors (contravariant tensors) by lower indexes and covectors (covariant tensors) by upper indexes because we often use indexes for exponents of degree.
Corresponding affine coordinates will be denoted by ξ, η: [ξ 0 : ξ 1 : Quite analogously, the conic B can be parameterized as . (4.5) Proof. The affine tangent line L to the curve A at a point Q with the parameter x has the direction vector τ = ( dξ dx , dη dx ). Therefore the affine point P satisfies the equality with some k ∈ R and an origin O. The last equality is equivalent to the complanarity of vectors E = (E 0 (x), E 1 (x), E 2 (x)), E ′ = (E ′ 0 (x), E ′ 1 (x), E ′ 2 (x)) and G = (G 0 (y), G 1 (y), G 2 (y)) in the bundle space R 3 \ {0} of the projective fiber bundle R 3 \ {0} → RP 2 , where the prime denotes the derivative with respect to x. Indeed, the collinearity of the vectors Return to our case of conics. We see F (x, y) has the form with polynomials M i (y) defined as where ǫ ikl is the completely antisymmetric tensor. One can easily check that deg(M i (x)) ≤ 2, therefore the curve F (x, y) = 0 is a biquadratic curve of the form (1.4). Note that the equality (4.6) can be written as F (x, y) = ( M (x), G(y) ) with the scalar product (·, ·) or F = M (x), G(y) with the pairing ·, · (in this case G ∈ R 3 , M ∈ (R 3 ) * ) and F = ( E(x), E ′ (x), G(y)) with the mixed product in R 3 . Further, we introduce vectors x = colon(1, x, x 2 ), y = colon(1, y, y 2 ) and matrixes E, G by the rules where the matrix A is obtained in the same way as the matrix E above, moreover A = (a ik ) with the matrix from (1.4). In a case of irreducible conics the matrix A is nondegenerate. Indeed, if the matrix A is degenerate then either the matrix M or the matrix G is degenerate by virtue of (4.8). The degeneracy of G means a linear dependence of polynomials G i (y) that is the conic B will be a straight line in this case. The degeneracy of M means that there exist constants c 0 , c 1 , c 2 such that i.e. a linear dependence of polynomials E i (x) and the conic A will be a straight line.
Return to the Poncelet construction. We have obtained the parameters x 1 of the point Q 1 and y 1 of the point P 1 satisfy the equation F (x 1 , y 1 ) = 0, with F from (4.5). Note now that instead the point P 1 we could write the point P 2 with the parameter y 2 in the Poncelet construction and have the same equation F (x 1 , y 2 ) = 0. We obtain the first result: for any point given by a parameter value x 1 on the conic A, the points with parameters y 1 and y 2 of the intersection points of the tangent line L 1 at x 1 with B are determined as two roots of the quadratic equation: F (x 1 , y) = 0. Thus identifying a point with its parameter value we can say the Poncelet mapping U B maps the point y 1 into the point y 2 and hence it (more precisely, the mapping I B ) coincides with the John mapping I 1 from section 2. It is analogously, the mapping U A maps the point x 1 into the point x 2 and hence it (the mapping I A ) coincides with the John mapping I 2 , and the mapping U coincides with the John mapping T on the curve C (1.4).
We have proved the following Proof. Let us have a decomposition (4.6) of a given biquadratic polynomial F . Then we have a parameterization of the conic B by means of projective coordinate G 0 , G 1 , G 2 and the first our aim is to find polynomials E 0 , E 1 , E 2 such that the equalities (4.7) hold. In other words, we must find a polynomial solution of the system of ordinary differential equations with a known vector M = (M 0 (x), M 1 (x), M 2 (x)), an unknown vector E = (E 0 (x), E 1 (x), E 2 (x)) and the vector product × in R 3 . One can interpret this equation as a problem to find a curve (more exactly, the tangent vector of a curve) if it is known its binormal. Now we will need the following Lemma 2. Consider the differential equation in R 3 with a known smooth vector function b = (b 1 (t), b 2 (t), b 3 (t)) and unknown vector function r = (r 1 (t), r 2 (t), r 3 (t)) depending on a real parameter t.  Proof. Note first that the Jacobi determinant of the system (4.10) of ordinary differential equations with respect to r ′ is identically equal to zero, so that a standard theory of differential equations systems does not work for this system. I). Let b × b ′ ≡ 0 on an interval. We observe that the solution must be only of the form r = v(t) b × b ′ because from (4.10) we easily obtain: b · r = 0, b · r ′ = 0, so b ′ · r = 0. The scalar v is unknown still. Substituting such r in (4.10) we obtain v 2 (b, b ′ , b ′′ )b = b so that there exists no solution of (4.10) when (b, b ′ , b ′′ ) < 0. The equality (b, b ′ , b ′′ ) = 0 on an interval gives b = 0, i.e. a contradiction. II). Let now b × b ′ ≡ 0 on an interval. If b ≡ 0 and b ′ ≡ 0 then b(t) = µ(t)b 0 with a scalar function µ, b 0 is a constant vector that means the curve r is a plane curve. Note that we examine simultaneously the case b ≡ 0 and b ′ ≡ 0 (µ ≡ 1). Substitute this b in (4.10) and choose another parametert such that r × r ′ = b 0 . It follows from this that r × r ′′ = 0, so r ′′ = ν(t)r with a scalar function ν. After a reparameterizationt → s we obtain r ′′ = ±r, i.e. r = r 0 e s + r 1 e −s or r = r 0 cos s+r 1 sin s with some constant vectors r 0 , r 1 of a plane which is orthogonal to b 0 . For r 0 = r 1 these solutions are cases of a hyperbolic and an elliptic rotation of the plane pespectively, besides, there are also cases of a movement along a direct straight line (cases b = 0 and r 0 = r 1 = 0) and of a stationary state.

4.4.
Projective invariance of the biquadratic curve. One can consider the vector E × E ′ as a covector E * of a dual space ( where the matrix A is from (4.2), more exactly, E * is proportional to E ×E ′ . Indeed, by definition we at once obtain E * (x), E(x) ≡ 0 and E * , E ′ ≡ 0. The covector field E * (x) describes a conic in the dual projective space, the field of tangent lines to A (see, e.g., [27]): Proof. In order to derive the first result let remember that an arbitrary nondenerate projective transformation P L of the affine plane can be obtained from a nondenerate linear transformation L of the bundle space R 3 \ {0} of the projective fiber bundle. Such linear transformation L gives a number factor det L for F in the formula (4.5).
Thus the arbitrary nondenerate projective transformation of the affine plane and conics A and B don't change the equation (1.4). Thus, the biquadratic curve F (x, y) = 0 depends only on a projective class of the conics pair. But the conics can be in one of the following generic dispositions (the order is chusen for a correspondence to proposition 15): 0) the conic B lies inside of A, strictly or not, i.e. no tangent line to A cuts B. 1) the conic A lies strictly (i.e. without contact) inside of B so that each tangent line to A cuts B in two different points.
2) the conic A cuts the conic B in four points and there exist a straight line that has no common point with A and B.
3) the conic A intersects the conic B at only two points without contacts. 4) the conic A lies strictly outside of B and the conic B lies outside of A. 5) the conic A cuts the conic B in four points and each straight line has a common point with A or B (there is no common tangent line, a hyperbola and an ellipse). Now we would like to clarify how one could know by F what case occurs. In order to clarify it let's take the following observations. If the conic A intersects B in a point P then from the point P ∈ B there is only a tangent line to A, so that in the Poncelet construction there is only a point (x, y) with the parameter y corresponding to P ∈ B that is the point (x, y) is an y-vertex of C, i.e. an extreme point along direction of real axis y. Remember that by the proposition 14 for generic cases 1)-5) the roots of D 2 (y 1 ) and only they are y-coordinates of y-vertices. Each such vertex corresponds to a point of intersection A ∩ B.
In the same way a common tangent line to conics A and B in the plane (ξ, η) gives us an x-vertex (x 2 , y 2 ) in the plane (x, y) and we have: if A 2 (x 2 ) = 0 then . And for generic cases 1)-5) real roots of D 1 (x 2 ) and only they are x-coordinates of x-vertices. Each such vertex corresponds to a common tangent line.
Let us remember that the statement 6 says the invariants g 2 and g 3 of polynomials D 1 and D 2 are the same. Note that the case ∆ < 0 (i.e. k 2 < 0) corresponds only to the above case 5) because the polynomial D 1 has two real and two complex roots if and only if ∆ < 0, therefore if and only if the polynomial D 1 is the same. We obtain the following Proposition 23. In the case of disposition 0) each of equations D 1 (x 1 ) = 0 and D 2 (y 1 ) = 0 has no real solution and the curve is not real. In the case 1) each of equations D 1 (x 1 ) = 0, D 2 (y 1 ) = 0 has no real solution, the curve C is real and it has no a vertex. In the case 2) each of equations D 1 (x 1 ) = 0, D 2 (y 1 ) = 0 has four real solutions, the curve C has four x-vertices and four y-vertices. In the case 3) each of equations D 1 (x 1 ) = 0, D 2 (y 1 ) = 0 has two real solutions, the curve C has two x-vertices and two y-vertices. In the case 4) we have four common points, so that the equation D 1 (x 1 ) = 0 has four real solutions, the equation D 2 (y 1 ) = 0 has no real solution, the curve C has no y-vertex and four x-vertices. In the case 5) we have four common tangent lines, so that the equation D 1 (x 1 ) = 0 has no real solution, the equation D 2 (y 1 ) = 0 has four real solutions, the curve C has no x-vertex and four y-vertices.
Any point of contact gives a point (x 1 , y 1 ) which is both x-vertex and y-vertex, i.e. D 1 (x 1 ) = D 2 (y 1 ) = 0, the conditions (3.18) of singular point are fulfilled, and according to the proposition 7 either the curve is irreducible or x 1 and y 1 are multiple zeros of the discriminantsD 1 (x) andD 2 (y) respectively and this singular point is unique. In both cases we observe either the case III of the John's list of dynamical system behaviors and a breakdown of smoothness of the curve or a case of degeneration. In this work we do not consider these cases. Some examinations are in the work [40].
Remark more that the statement 7 and proposition 15 of the subsection 3.4 imply Proposition 24. For any two conics that are in one of generic dispositions 1)-5) there are real linear-fractional changes R 1 , R 2 of real parameters x = R 1 (x), y = R 2 (ȳ) such that the corresponding reduced biquadratic curve C will be having one of the canonical form Note that last proposition implies the statement of the big Poncelet theorem. Since the Poncelet problem is invariant under the arbitrary projective transformation of the plane (ξ, η) we can reduce the conics A and B to some simple shapes. Consider several possibilities.
(i) If we reduce A and B to concentric quadrics determined by the equations In this case we have the parameterization ξ = 2a for the conic A and ξ = 2a for the conic B. It is easily verified that the polynomial Z(x, y) defined by (4.5) is reduced in this case to the simplest Euler-Baxter form: Z = x 2 y 2 + 1 + a(x 2 + y 2 ) + 2bxy with complex coefficients.
Hence, we have a simple form solution of the Poncelet problem in this case as above (see (3.39)): Parameters a i and b i can still be specialized further. For instance, it is possible to choose a 2 = b 2 = 1 reducing B to the unit circle. This choice corresponds to so-called Bertrand model of the Poncelet process [50]. If the second conic is an ellipse ξ 2 /a 2 1 + η 2 /b 2 1 = 1, a 1 > 1, b 1 < 1 then the formula (4.5) gives us a bounded curve of the form xy + 1 = 0. (4.14) In this case the points x n are isomorphic to the godograph distribution of spins in the classical XY -chain (see section 7.1 and [26]). Another possible choice a 2 − a 1 = b 2 − b 1 corresponds to the confocal quadrics. In this case the Poncelet problem is equivalent to the elastic billiard [58], [41].
(ii) Let us consider a possibility to reduce the conics B and A to two parabolas in the euclidean plane (ξ, η). One of them, say B, can be fixed by the choice ξ = x, η = x 2 , whereas another parabola remains arbitrary: ξ = F 1 (y)/F 0 (y), η = F 2 (y)/F 0 (y), where F 0 (y) = (ay + b) 2 is a square of the linear function (this is a characteristic property of any parabola). Then we get Performing additional projective (complex) transformation of the variable y we can fix polynomials F 1 (y), F 2 (y), and F 0 (y) in such a way that the polynomial Z(x, y) becomes symmetric in x, y. Then we can reduce Z(x, y) to the form Z = (xy + (x + y)y 0 + g 2 /4) 2 − (x + y + y 0 )(4xyy 0 − g 3 ), (4.16) where g 2 , g 3 are two remaining (arbitrary) independent parameters of the polynomial Z.
As we will see in Sect. 7, the biquadratic curve of similar form appears for the phase portrait of the elliptic solution of the Toda chain.
Finally, we note that when F 0 = const and F 1 (y) is a linear function in y then the parabola A has its axis parallel to that of the parabola B. But then the curve Z(x, y) = 0 describes arbitrary conics in the coordinates x and y with absent x 2 y 2 , x 2 y, and xy 2 terms. 4.6. Cayley determinant criterion. Let A, B be arbitrary conics as described in previous section. Recall, that all tangents are passing through points of the conic A, whereas all vertices lie on conic B. (It is possible to assume that A is located inside the conic B). Let M A and M B are 3 × 3 matrices describing conics (i.e. corresponding quadratic forms) in projective coordinates x 0 , x 1 , x 2 . I.e. if the conic A is the unit circle x 2 + y 2 = 1 and conic B is concentirc circle with the radius R then quadratic forms for A, B are Corresponding matrices M A , M B are diagonal: Compute the characteristic determinant F (z) = c 0 + c 1 z + c 2 z + · · · + c n z n + . . . (4.19) Then we compute the Hankel-type determinants: Then the Cayley criterion [13], [28] is: Here the degree of R is even: deg R = 2m, deg ρ = m − 1 and ρ/A = 2P ′ /Q. A main point of Abel's considerations is that the polynomials P and Q satisfy the equation (5.1) with L = 1. And conversely, if the polynomials P and Q satisfy the equation (5.1) with L = 1 then the equality (5.2) holds with ρ = 2P ′ /Q and A = 1. Thus, the solvability of the Pell-Abel equation plays a role a integrability criterion of Abel differential. It is well-known that later J. Liouville, V.V. Golubev and successors shown if the integral in the left-side part of (5.2) with some ρ and R can be written by elementary functions then the right-side part must have the view (5.2) with some P and Q and the same ρ. Note also, Abel gave one more criterion for the representation (5.2), namely the formula (5.2) holds iff polynomial continued fraction of the function √ R is periodic. Thus, the solvability of the Pell-Abel equation plays also a role of a criterion of periodicity of continued fraction.
Consider one more classical problem -Chebyshev's problem on searching of a polynomial of least deviation. Let us have a system of l closed intervals of real axe (a j , b j ) and the following problem. One should find a polynomial of given degree n with unit leading coefficient that has a least deviation on the set I, i.e. find a minimum of the functional t n − P n−1 (t) C(I) . A general polynomial P n−1 (t) runs a finite-dimensional lineal space and the problem can be interpreted as a problem to find an element in a finite-dimensional lineal subspace of a Banach space which is nearest to a given. But because the space C(I) isn't reflexive such nearest element isn't obliged to exist.
By a method going back to P.L. Chebyshev, one succeeded in proving the existence of such minimal polynomial [2]. Further, if the polynomial P n−1 is minimal on the set I then, possible, it will also be minimal on some greater closed n-right extension of the set I [51]. If now one take a polynomial R in the form then it is proved to be that the solvability of the Pell-Abel equation (5.1) with unknowns P (t), Q(t), L = const > 0 is equivalent to that the set E is n-right extension of the set I. In addition the polynomial P gives a solution of extremal problem, and the number √ L is a least deviation (minimum of the norm) [51].
It is interesting that in this case the subset E is continuous spectrum (which is absolutely continuous and two-valued) of some bi-infinite Jacobi (3-diagonal) selfajoint real periodic matrix in the space l 2 if and only if the set E is n-right or that is equivalently if the Pell-Abel equation (5.1) is solvable (see references in [51]). Remark that the Chebyshev's problem will be got for the case of one interval E = [−1, 1], R = (1 − t 2 ), and the Akhiezer's problem -for the case of two intervals (deg R = 4) and polynomial of the fourth order . Note that there are several solvability criterions for the Pell-Abel equation with a polynomial corresponding to the Akhiezer's problem, among them is well-known Zolotarev's porcupine (see, for instance, [43]). Note also recent Khrushchev's work [39] on links to continued fractions.

5.2.
Connection between the Poncelet problem and the Pell-Abel equation. The solvability of the Pell-Abel equation with the polynomial R of even order, as we have noted above, has several equivalent formulations ( [51]). In the work of V.A. Malyshev [43] there is a new solvability criterion given in an algebraic form that we will formulate for interesting for us case of the fourth order R = t 4 + d 1 t 3 + ... + d 4 . Let us expand the root √ R in Laurent series in a neighborhood of infinity: and make up a determinant of Hankel type: Our observation is following. Let consider the Pell-Abel equation (5.3) with R of the fourth order and let λ 1 be one of roots of polynomial R(t). Make the shift of parameter t → t + λ 1 (i.e. t =t + λ 1 ). New Pell-Abel equation (5.3) will also be solvable. Now apply the Malyshev's criterion to new equation and let the coefficients C j be coefficients of Laurent series of the root R(t + λ 1 ). Then the equality Γ k = 0 is necessary and sufficient for solvability of the equation (5.3) with given orders of polynomials P, Q. On the other hand, let in initial setting make change of variable s = 1/(t − λ 1 ), t = λ 1 + 1/s. Then we obtain with a polynomial F (s) of the third order and therefore √ R = F (s)/s 2 . Note that the polynomial R can be restored from F by inverse transformation: where z 1 , z 2 , z 3 − roots of the polynomial F of the third order. The polynomial F can be generated by formula F = det(A − zB) and matrixes A = diag(z 1 , z 2 , z 3 ), B = diag(1, 1, −1) that are built by quadratic forms The expansion (4.19) of the root F (s) gives us the expansion F (s)/s 2 = c 0 /s 2 + c 1 /s + c 2 + · · · + c n s n−2 + . . . which after inverse change can be written as R(t + λ 1 ) = c 0 t 2 + c 1 t + c 2 + c 3 /t + · · · + c n /t n−2 + . . . .

Criterions of uniqueness breakdown for the Dirichlet problem and Ritt's problem
In this section we give criterions of solution uniqueness for the Dirichlet problem (1.1),(1.2) and show how to construct a system of some special solutions in case of non-unique the Dirichlet problem for the string equation. We describe a method that can be applied to a slightly more general class of curves than biquadratics.
We'll consider the Dirichlet problem (1.1), (1.2) in the classical setting (2.2) if the curve C is bounded and in modified setting (2.3) if the curve C is unbounded because we'll need an application of the proposition 1.
Remind that in subsection 3.5 we consider an even periodic meromorphic function φ(z) with some (complex) period T and a complex curveC ⊂ C 2 described parametrically as x(t) = φ(t), y(t) = φ(t + ε), t ∈ C (6.1) with the property (3.35), ε is a nonzero complex parameter. We considered also a real curve C =C ∩ R 2 given by means of a contraction of the functions x, y on a set S ∈ C. We saw that the curveC is symmetric with respect to the line y = x, the complex John mapping T is equivalent to a shift of the parameter t by the step −2ε and the periodicity condition of the complex John mapping has the form: 2nǫ = mT with some integer n, m.
We assume also (this is a very strong restriction, as we will soon see) that the function φ(z) possesses a nontrivial multiplication property: where R n (z) is a family of rational functions of the argument z (by definition, R 1 (z) = z, but for other n = 2, 3, . . . the expression and even degree of R n (z) can be non-obvious). We will consider the condition Consider the complex setting (2.7) of homogeneous Dirichlet problem. Now the sufficient condition of the uniqueness of the Dirichlet problem from proposition 2 requires the transitivity of T that implies that for each integers n and m we have 2nε = mT .
Assume now that this condition isn't fulfilled, i.e. for some integers N, M we have the condition 2N ε = M T (6.4) We show that in this case the problem is indeed non-unique and we construct explicitly a system of explicit solutions Φ n (x, y) = f n (x) + g n (y), n = 1, 2, . . . (6.5) for the string equation in the domain bounded by the curve C.
As the rational functions R n (z) are assumed to be non zero, bounded on R 2 and non-constant then the function Φ(x, y) = f n (x) + g n (y) = R 2nN (x) − R 2nN (y) is obviously nonzero and smooth in R 2 . So we should verify the Dirichlet boundary condition Φ(x, y) ≡ 0 in all points of the curve C. Indeed, for any point t in the curve we have where we exploited properties (3.34), (6.2) and (6.4). Hence the function Φ n (x, y) is identically zero in all points on the curve C.
We proved the following Proposition 28. Under the condition (6.4) for the curve C we can choose f n (x) = R 2nN (x), g n (y) = −R 2nN (y), (6.6) where R n (z) are rational functions defined by (6.2) with property (6.3). This will provide a non-zero solution (6.5) of the Dirichlet problem in the complex setting (2.7). If, in addition, the condition (6.3) is fulfilled then obtained solution satisfies the modificed setting (2.3) and if furthermore the curve is bounded then we have a classical solution in interior of the curve C, i.e. in the setting (2.2).
We first illustrate how this theorem works in simplest case when the curve C is ellipse. In this case we can choose parameterization x(t) = cos(t), y(t) = cos(t + ε) (6.7) with some parameter ε. Note that the ellipse described by (6.7) has semiaxis a = √ 2 cos(ε/2), b = √ 2 sin(ε/2) one of which is inclined by the angle π/4 with respect to axis x. Thus, in case of nonuniqueness of the Dirichlet problem for ellipse we get a denumerable family of polynomial nonzero solutions inside the ellipse.
Let C be the biquadratic curve. In beginning, let's consider the case (3.42) of subsection 3.5: a > 0, c = −1. Here the curve is parameterized as ). An easy analysis shows the last curve will be real for real t and real η. For the elliptic cosine there are well-known multiplication formulae cn(2mz) = T 1 m (cn 2 z), cn((2m + 1)z) = cn z T 2 m (cn 2 z) with some rational functions T 1 n (z), T 2 n (z). Poles of the function cn(z) lie on the line Im t = K ′ therefore the condition (6.3) holds, the curve C is bounded in this case and the proposition 28 states the condition (3.49) is sufficient for uniqueness breakdown for the Dirichlet problem in classical setting. Necessity of the condition (3.49) follows from the proposition 1 (for details see [56]).
There are also 3 exceptional cases connected with elliptic functions with certain restrictions upon their parameters: (iii) φ(z) = a℘ 2 (z)+b c℘ 2 (z)+d with arbitrary a, b, c, d but with restriction g 3 = 0 and g 2 arbitrary real. In this case the fundamental parallelogram of periods 2ω 2 , 2ω 3 is a simple square. Such case corresponds to so-called lemniscatic elliptic functions.
(iv) φ(z) = a℘ 3 (z)+b c℘ 3 (z)+d with arbitrary a, b, c, d but with restriction g 2 = 0 and g 3 arbitrary real. In this case the period lattice has the hexagonal symmetry. Such case corresponds to so-called equianharmonic elliptic functions.
For the equianharmonic case Ritt found one more case when φ(z) is a linear rational combination of the Weierstrass function ℘ ′ (z), but in this case the function φ(z) is not even and we can omit this case.
The case (ii) corresponds to our problem in biquadratic. Thus, in the case of nonuniqueness we have the solution f n (x) = R 2nN (x), g n (y) = −R 2nN (y), (6.11) where R n (z) are rational functions defined as (6.2). Note that in contrast to the trigonometric case (i.e. φ(z) = cos(z)) there are no simple explicit expressions for R n (z) for arbitrary n.
What about cases (iii) and (iv)? In these cases we have that the curve C is described by an algebraic equation of degree greater than 4. Consider, e.g. the case (iii) and assume for simplicity that φ(z) = ℘ 2 (z). We are seeking an algebraic equation f (x, y) = 0 for the curve C described parametrically as x(t) = ℘ 2 (t), y(t) = ℘ 2 (t + ǫ). This equation can be easily found from addition theorem for the Weierstrass function. We will not write down it explicitly, let us mention only that degree of this equation is 8. Analogously, for the case (iv) we obtain the algebraic curve describing by a polynomial F (x, y) of a total degree 12. But we should note that the John condition of intersection at no more than two points with vertical and horizontal lines is not satisfying and here we do not have a sufficient condition of solution uniqueness for boundary value problems.
7. Related problems of mathematical physics 7.1. Classical Heisenberg XY spin chain. There is an interesting relation with the classical Heisenberg XY spin chain [26] which is a system of 2-dimensional unit vectors ("spins") r n = (x n ; y n ), | r n | = 1 with energy of interaction ( r n , J r n+1 ), (7.1) where J = diag(J 1 , J 2 ) is a diagonal 2×2 matrix. The problem is to find static solutions providing local extremum to the energy E. As was shown in [26] this problem is equivalent to finding solutions of the systems of non-linear vector equations: ( r n , J( r n−1 + r n+1 )) = 0, n = 1, 2, . . . , N − 1.
Below we would like to select cases of closed chain when r 0 = r N for some N . In what follows we will assume that J 1 = 1, J 2 = j > 1. Among all solutions of (7.2) we choose so-called regular ones [26] with the condition Then it is possible to show that the scalar product W = ( r n , J r n+1 ) (7.4) doesn't depend on n and hence can be considered as an integral of the system (7.2). It is enough to construct general regular solutions [26] which are of two types. Choice of solution depend on the value of integral W . If |W | < 1/j then x n = cn(q(n − θ); k), y n = sn(q(n − θ); k), (7.5) where parameters k, q can be found from If 1/j < |W | < 1 then x n = dn(q(n − θ); k), y n = k sn(q(n − θ); k), where cn(q; k) = 1/j, In both cases parameter θ is arbitrary real number depending on initial condition. If the chain is periodic then we have qN = 4Km 1 + 2iK ′ m 2 (7.9) which coincide with (3.40). The reason for such coincidence is the following. Consider the equation of the integral x n x n+1 + j −1 y n y n+1 = W (7.10) with fixed value W . The equation (7.10) can be reduced to the algebraic form (3.28) by standard substitution (stereographic projection from the unit circle to the line): x n = 1 − u 2 n 1 + u 2 n , y n = 2u n 1 + u 2 n It is easily seen that the variables u n , u n+1 lie on the Euler-Baxter biquadratic curve u 2 n u 2 n+1 + 1 + a(u 2 n + u 2 n+1 ) + bu n u n+1 = 0 with parameters a, b simply related to the "physical" parameters j, W .
Then it is easily verified that finding solutions (step-by-step) of equations (7.2) for the regular solutions are equivalent to finding points M 2 , M 3 , . . . M N for the John algorithm. Note that arguments similar to ones in the formula (3.41) give us that the parameters q and θ must be real so that the condition (7.9) takes the form q 4K = m 1 N . (7.11) Thus static regular solutions for the closed finite classical Heisenberg XY -chain are equivalent to periodic solutions of the John algorithm for the Euler-Baxter biquadratic curve, or, equivalently, to finding periodicity condition of the Poncelet process for the unit circle x 2 + y 2 = 1 and a concentric conic x 2 /ξ 1 + y 2 /ξ 2 = 1. This choice of conics corresponds to the Bertrand model of the Poncelet process [50]. This means that there is an equivalence between static periodic solutions of the Heisenberg XY -chain and the Bertrand model of the Poncelet process. 7.2. Elliptic solutions of the Toda chain and biquadratic curve. Now we consider the Toda chain that is a discrete dynamical system consisting of two sets u n (t), b n (t) of complex variables depending on continuous parameter t and discrete parameter n = 0, ±1, ±2, . . . . The equations of the motion are [57] b n = u n+1 − u n ,u n = u n (b n − b n−1 ) (7.12) The Toda chain is one of the most simple and famous completely integrable discrete dynamical systems (for history of this model and review of different approaches see [57]). Among numerous explicit solutions there are so-called "elliptic waves" constructed firstly by Toda himself [57]. We give the Toda elliptic solutions in somewhat different form which is more convenient for applications (Toda used Jacobi elliptic functions whereas we exploit the Weierstrass functions).
Proof. The Toda chain equations (7.12) are verified directly by substitution using well known formulas [60]: d dz ζ(z) = −℘(z), d dz lg σ(z) = ζ(z), ℘(x) − ℘(y) = σ(y − x)σ(x + y) σ 2 (x)σ 2 (y) Strictly speaking, the variable b n (t) is inessential -it can be eliminated from the system (7.12). Thus only variable u n (t) is considered as a "true" Toda chain variable (for details see [57]). Now we can construct the phase portrait for this variable. By the phase portrait we will assume the plot constructed on the plane x, y if one take points P 0 , P 1 , P 2 , . . . with the coordinates P 0 = (u 0 , u 1 ), P 1 = (u 1 , u 2 ), . . . , P n = (u n , u n+1 ). The phase portrait is an indicator of integrability: if the system is integrable the points P i , i = 0, 1, . . . fill some smooth curve. Otherwise these points are distributed quasi-stochastically (so-called "stochastic web" [21]). In our case the variable u n (t) is given explicitly by (7.13), u n+1 (t) = u n (t − p/ω) and hence the points P n = (u n , u n+1 ) fill the symmetric biquadratic curve in its canonical form (3.24) by the proposition 9. The complex John algorithm in this biquadratic curve is equivalent to passing from the point P n to point P n+1 and then to point P n+2 . Note that the time parameter t describes a smooth motion along this curve (Hamiltonian dynamics), whereas the John algorithm describes a "discrete motion". The periodic case is defined by a positive integer N such that u n (t) = u n+N (t). From (7.13) it is seen that periodicity condition is equivalent to pN = 2m 1 ω 1 + 2m 2 ω 2 (7.14) with some integers m 1 , m 2 . An important property of the elliptic Toda solutions is that the periodicity property (7.14) doesn't depend on the time parameter t, i.e. if periodic condition takes place for one value of t then it also holds for all others values of t. In terms of the (complex) John algorithm this means that a period of a point following the John mapping doesn't depend on choice of this initial point P 0 = (u 0 (t), u 1 (t)) on the curve. Comparing obtained formulas for u n (t) with (4.15) and (4.16) we can say that the elliptic solutions of the Toda chain give an interesting illustration of the Poncelet theorem: periodic solutions (7.14) of the Toda chain correspond to periodicity of the Poncelet process on two parabolas. 7.3. Elliptic grids in the theory of the rational interpolation. In this section we describe briefly an interesting connection of the Poncelet problem with theory od so-called admissible grids for biorthogonal rational functions. This subject is important in theory of rational interpolations. For further details and relations with theory of special functions see, e.g. [42], [52], [53], [54], [63], [64].
Let us consider the set of rational functions R n (x) of the order [n/n], which means that R n (x) are given by ratios of two n-th degree polynomials in x. In what follows it is assumed that all rational functions R n (x) have only simple poles.
We take α 1 , α 2 , . . . , α n as n distinct prescribed positions of the poles of R n (x). Then R n (x) can be written as a sum of partial fractions with the coefficients t (n) i , i = 1, 2, . . . , n, playing the role of residues of R n (x) at the poles α i . The coefficient t (n) 0 can be interpreted as lim x→∞ R n (x). Let x(s) be a "grid", i.e. a function in the argument s. We would like to construct so-called lowering operator D x(s) in the space of rational functions R n (x) defined on this grid in the following way. We take as a definition of the lowering operator D x(s) a divided difference operator in the parameterizing variable s, which obeys the following properties: i) the grid x(s) is a meromorphic function of s ∈ C which is invertible in some domain of the complex plane; (ii) for any function F (s) one has D x(s) F (s) = χ(s)(F (s + 1) − F (s)), where χ(s) is some function to be determined; (iii) D x(s) R 1 (x) = const, where R 1 (x) is an arbitrary rational function of the order [1/1] with the only pole at x = α 1 ; (iv) the operator D x(s) transforms any rational function R n (x) with the prescribed poles α 1 , . . . , α n to a rational functionR n−1 (y(s)) of the adjacent grid y(s) with some other poles β 1 , . . . , β n−1 ; (v) the operator D x(s) is "transitive": for any nonnegative integer j the operator D (j) x(s) defined as D (j) x(s) F (s) = χ j (s)(F (s + 1) − F (s)) with some function χ j (s) transforms any rational function R n (x) with the prescribed poles α j+1 , . . . , α j+n to a rational functionR n−1 (y(s)) with one and the same adjacent grid y(s) and the sequence of poles β j+1 , . . . , β j+n−1 ; (vi) we assume that the poles are nondegenerate: there are infinitely many distinct values of α n and β n and α n = α n+1 , α n+2 and, similarly, β n = β n+1 , β n+2 for all n.
An important restriction is the condition of independence of D x(s) on the order n of a rational function. The problem is to deduce the functions χ j (s) and x(s), y(s) from the given set of requirements.
From the properties (i)-(iii) we easily find The function χ(s) is defined up to an inessential constant multiplier. Note that from (ii) we have D x(s) R 0 (z) = 0. The most non-trivial problem in the construction of the operator D x(s) consists in establishing the properties (iv)-(v).
This problem was solved in [54]. It appears that the grid x(s) as well as the grids y(s), α s , β s should belong to the class of the elliptic grids. This means that they should satisfy the biquadratic equation A 1 x 2 (s + 1)x 2 (s) + A 2 x 2 (s)x(s + 1) + A 3 x 2 (s + 1)x(s) + A 4 x(s + 1)x(s) + A 5 x 2 (s + 1) + A 6 x 2 (s) + A 7 x(s + 1) + A 8 x(s) + A 9 = 0 (7.17) with some constants A i , i = 1, 2, . . . , 9. As we already know this equation can be parameterized in terms of the elliptic functions and we thus arrive at elliptic grids x(s). Thus the elliptic grids x(s), y(s) are the most general ones to provide existence of the lowering operator for rational functions.
In the theory of the rational (sometimes called the Cauchy-Jacobi or Padé) interpolation these grids appear naturally for some class of self-similar solutions.
The Cauchy-Jacobi interpolation problem (CJIP) for the sequence Y j of (nonzero) complex numbers can be formulated as follows [8], [44]. Given two nonnegative integers n, m, choose a system of (distinct) points x j , j = 0, 1, . . . , n + m on the complex plane. We are seeking polynomials Q m (x; n), P n (x; m) of degrees m and n correspondingly such that Y j = Q m (x j ; n) P n (x j ; m) , j = 0, 1, . . . n + m (7.18) (in our notation we stress, e.g. that polynomial Q m (x; n), being degree m in x, depends on n as a parameter). It can happens that solution of the CJIP doesn't exist. In this case it is reasonable to consider a modified CJIP: Y j P n (x j ; m) − Q m (x j ; n) = 0, j = 0, 1, . . . , n + m, (7.19) where polynomials P n (x; m), Q m (x; n) can be now unrestricted. The problem (7.19) always has a nontrivial solution. In exceptional case, if the system (7.18) has no solutions, some zeroes of polynomials P n (z; m) and Q m (z; n) coincide with interpolated points x s . Such points, in this case, are called unattainable [44].
The CJIP is called normal if polynomials Q m (x; n), P n (x; m) exist for all values of m, n = 0, 1, . . . and polynomials Q m (x; n), P n (x; m) have no common zeroes. This means, in particulary, that polynomials Q m (x; n), P n (x; m) have no roots, coinciding with interpolation points, i.e. Q m (x j ; n) = 0, P n (x j ; m) = 0, j = 0, 1, . . . n + m (7.20) In a special case when there exists an analytic function f (z) of complex variable such that f (x j ) = Y j the corresponding CJIP is called multipoint Padé approximation problem [8].
It is possible to show that the Cauchy-Jacobi interpolation problem is equivalent to theory of biorthogonal rational functions [64]. The elliptic solutions (obtained first in [52]) of this problem appear naturally if the interpolated function f (z) satisfy the so-called discrete Riccati equation [42]. Geometric interpretation of obtained elliptic grids and their connection with the Poncelet problem can be found in [42] and [54].
Note that if terms of degree > 2 are absent in (7.17) (i.e. A 1 = A 2 = A 3 = 0) then corresponding grid is degenerated to so-called Askey-Wilson grid [7] which is the most general grid for orthogonal polynomials satisfying a linear second-order difference equation [61]. From geometrical point of view the Askey-Wilson grids correspond to the John algorithm for the second-degree curves (i.e. ellipsis, hyperbola or parabola) [42].