Geometric Aspects of the Painlev\'e Equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$

The Riemann-Hilbert approach for the equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlev\'e varieties, the Painlev\'e property, special solutions and explicit B\"acklund transformations.


Introduction
The aim of this paper is a study of the Painlevé equations PIII(D 6 ) and PIII(D 7 ) by means of isomonodromic families in a moduli space of connections of rank two on the projective line. This contrasts the work of Okamoto et al. on these third Painlevé equations, where the Hamiltonians are the main tool. However, there is a close relation between the two points of view, since the moduli spaces turn out to be the Okamoto-Painlevé varieties. We refer only to a few items of the extensive literature on Okamoto-Painlevé varieties. More details on Stokes matrices and the analytic classification of singularities can be found in [11].
A rough sketch of this Riemann-Hilbert method is as follows (see for some background [10] and see for details concerning PI, PII, PIV [8,9,12]). The starting point is a family S of differential modules M of dimension 2 over C(z) with prescribed singularities at fixed points of P 1 . The type of singularities gives rise to a monodromy set R built out of ordinary monodromy, Stokes matrices and 'links'. The map S → R associates to each module M ∈ S its monodromy data in R. The fibers of S → R are parametrized by T ∼ = C * and there results a bijection S → R × T . The set S has a priori no structure of algebraic variety. A moduli space M over C, whose set of closed points consists of certain connections of rank two on the projective line, is constructed such that S coincides with M(C). There results an analytic Riemann-Hilbert morphism RH : M → R. The fibers of RH are the isomonodromic families which give rise to solutions of the corresponding Painlevé equation. The extended Riemann-Hilbert morphism RH + : M → R × T is an analytic isomorphism. From these constructions the Painlevé property for the corresponding Painlevé equation follows and the moduli space M is identified with an Okamoto-Painlevé space. Special properties of solutions of the Painlevé equations, such as special solutions, Bäcklund transformations etc., are derived from special points of R and the natural automorphisms of S.
The above sketch needs many subtle refinements. One has to construct a geometric monodromy spaceR (depending on the parameters of the Painlevé equation) which is a geometric quotient of the monodromy data. The 'link' involves (multi)summation and in order to avoid the singular directions one has to replace T by its universal coveringT ∼ = C. In the construction of M, one represents a differential module M ∈ S by a connection on a fixed vector bundle of rank two on P 1 . This works well for the case (1, −, 1/2), described in Section 2. In case (1, −, 1), described in Section 3, this excludes a certain set of reducible modules M ∈ S. The moduli space of connections M is replaced by the topological coveringM = M × TT . Now the main result is that the extended Riemann-Hilbert morphism RH + :M →R ×T is a well defined analytic isomorphism.
Each family of linear differential modules and each Painlevé equation has its own story. For the family (1, −, 1/2), corresponding to PIII(D 7 ), the computations of R, M and the Bäcklund transformations present no problems. For the resonant case (i.e., α = ±1 and θ ∈ Z, see Sections 2.1, 2.2 and 2.5 for notation and the statement) there are algebraic solutions of PIII(D 7 ). The spaces R(α) with α = ±i have a special point corresponding to trivial Stokes matrices. This leads to a special solution q for PIII(D 7 ) and θ ∈ 1 2 + Z, which is transcendental according to [5]. According to [2], q is a univalent function of t and is a meromorphic at t = 0.
For the general case (i.e., α = β ±1 ) of the family (1, −, 1), corresponding to PIII(D 6 ), the computations of R, M present no problems. The formulas for the Bäcklund transformations, derived from the automorphisms of S, have denominators. These originate from the complicated cases α = β ±1 and/or α = ±1 where reducible connections and/or resonance occur. Isomonodromy for reducible connections produces Riccati solutions and resonance is related to algebraic solutions. 2 The family (1, −, 1/2) and PIII(D 7 ) In this section the set S consists of the (equivalence classes of the) pairs (M, t) of type (1, −, 1/2) (see [10] for the terminology), corresponding to the Painlevé equation PIII(D 7 ). The differential module M is given by dim M = 2, the second exterior power of M is trivial, M has two singular points 0 and ∞. The Katz invariant r(0) = 1 and the generalized eigenvalues at 0 are normalized to ± t 2 z −1 with t ∈ T = C * . The singular point ∞ has Katz invariant r(∞) = 1/2 and generalized eigenvalues ±z 1/2 . Further (M, t) is equivalent to (M , t ) if M is isomorphic to M and t = t .
The Riemann-Hilbert approach to PIII(D 7 ) in this section differs from [10] in several ways. The choice (1, −, 1/2) is made to obtain the classical formula for the Painlevé equation. Further we consider pairs (M, t) rather than modules M . This is needed in order to distinguish the two generalized eigenvalues at z = 0 and to obtain a good monodromy space R. Finally, the definitions of the topological monodromy and the 'link' need special attention.

The construction of the monodromy space R → P
This is rather subtle and we provide here the details. Given is some (M, t) ∈ S and we write δ M for the differential operator on M . First we fix an isomorphism φ : up to a transformation (E 1 , E 2 ) → (µE 1 , µE 2 ) with µ ∈ C * . We require that φ(E 1 ∧ E 2 ) = 1 and then µ ∈ {1, −1}. The solution space V (∞) at z = ∞ has basis The formal monodromy and the Stokes matrix have on the basis e 1 , e 2 the matrices 0 −i −i 0 , 1 0 e 1 . Their product (in this order) is the topological monodromy top ∞ at z = ∞.
3. The link L : V (0) → V (∞) is a linear map obtained from multisummation at z = 0, analytic continuation along a path from 0 to ∞ and the inverse of multisummation at z = ∞.
The matrix 1 2 3 4 of L with respect to the bases e 1 , e 2 and f 1 , f 2 has determinant 1, due to the isomorphism φ : Λ 2 M → (C(z), z d dz ) and the careful choices of the bases. The relation α αc 2 One observes that α, c 1 , c 2 are determined by L and top ∞ . Thus the affine space, given by the above data and relations, has coordinate ring 4. The group G generated by the base changes (e 1 , e 2 ) →(−e 1 , −e 2 ) and (f 1 , f 2 ) →(λf 1 , λ −1 f 2 ), acts on this affine space. The monodromy space R is the categorical quotient by G and has coordinate ring This is in fact a geometric quotient. The morphism R → P = C * is given by (e, 12 , 14 , 23 , 34 ) → α := −i 14 e + i 12 − i 34 = 0. For a suitable linear change of the variables, the fibers R(α) of R → P are nonsingular, affine cubic surfaces with three lines at infinity, given by the equation A detailed calculation resulting in this equation is presented in [10,Section 3.5].
then the fiber is one point. If x 1 x 2 = 0, then the fiber is an affine line. Since C 2 \ S is simply connected, R(α) \ L is simply connected. Then R(α) is simply connected, too.
Remark on the differential Galois group. The differential Galois group of a module M , with (M, t) ∈ S, can be considered as algebraic subgroup of GL(V (0)). It is the smallest algebraic subgroup containing the local differential Galois group G 0 ⊂ GL(V (0)) at z = 0 and L −1 G ∞ L, where G ∞ ⊂ GL(V (∞)) is the local differential Galois group at z = ∞. Now G 0 is generated (as algebraic group) by the exponential torus s 1 0 0 s −1 1 s 1 ∈ C * , the formal monodromy α 0 0 α −1 and the Stokes maps The group G ∞ is (as algebraic group) generated by the exponential torus s 2 0 0 s −1 2 s 2 ∈ C * , the formal monodromy 0 −i −i 0 and the Stokes map 1 0 e 1 . This easily implies that the differential Galois group is SL(2, C).
In particular, M is irreducible and the same holds for the differential module C( m √ z) ⊗ M over C( m √ z) for any m ≥ 2. The construction needed to define the topological monodromy and the link. For the definition of the link and the topological monodromies we have to choose nonsingular directions for the two multisummations and a path from 0 to ∞. At z = ∞ the singular direction does not depend on t ∈ T and we can take a fixed nonsingular direction. However, at z = 0, the singular directions for t ∈ T = C * , t = |t|e iφ are φ and π + φ and they vary with t. Thus we cannot use a fixed path from 0 to ∞. In order to obtain a globally defined map L : 1+r on the universal covering of P 1 \ {0, ∞}. Now L is defined by summation at z = 0 in the direction φ − π 2 , followed by analytic continuation along the above path and finally the inverse of the summation at z = ∞ in the direction π 2 . WriteS = S × TT . The elements ofS are the pairs (M,t) with (M, et) ∈ S. For the elements inS the link and the monodromy at z = 0 are defined as above. Since R is a geometric quotient, [10, Theorem 1.9] implies: The above mapS → R ×T is bijective. The mapS(α) → R(α) ×T is bijective, sinceS → R ×T is bijective. . It follows that Λ 2 (V, ∇) is isomorphic to The condition at z = 0 is a 2 + bc ∈ tz −1 +θ The condition at z = ∞ is a 2 + a + bc = z + C[[z −1 ]], equivalently The space, given by the above variables and relations has to be divided by the action of the group {e 1 → e 1 , e 2 → λe 2 + (x 0 + x 1 z)e 1 } (with λ ∈ C * , x 0 , x 1 ∈ C) of automorphisms of the vector bundle. Using the standard forms below one sees that this is a good geometric quotient.
By gluing the two standard forms, one obtains the nonsingular moduli space M(θ). The map M(θ) → S(α), where α = e iθ , is a bijection.
Observation. After scaling some variables one sees that M(θ) is the union of two open affine spaces U 1 × T and U 2 × T , where U 1 is given by the variables a 1 , b 1 , c 0 and the relation a 2 1 + (1 − a 1 − b 1 c 0 )c 0 = 0, and U 2 is given by the variables a −1 , b 0 , c −1 and the relation Using the two projections The space M (θ) is simply connected since it is the union of the two simply connected spaces U 1 , U 2 .
Comment. The existence of an analytic isomorphism as in Theorem 2.2 is called the "geometric Painlevé property" in [1]. They prove this property for a number of Painlevé equations under a restriction on the parameters (loc. cit., Theorem 6.3). We prove it here for PIII(D 7 ) and in Sections 3.3.2 and 3.4.4 below for PIII(D 6 ) without any restriction.

Isomonodromy and the Okamoto-Painlevé space
The calculation is done on the 'chart' c 0 = 0 and q is supposed to be invertible. The data for the operator z d dz + A are for * = H, 1, 2 and B * ,i only depending on t. Using the Lie algebra structure one obtains the equations: By Maple one obtains the system and finally We note that the change q = −Q, t = −T brings this equation in the form * * This is the standard form for PIII (D 7 ) (see [6]). As in [8,9,12] one obtains: has the Painlevé property. The analytic fibrationt : is the space of initial values.

Automorphisms of S and Bäcklund transforms
The automorphism s 1 of S is defined by s 1 (M, t) = (M, −t). The induced action on R leaves all data invariant except for interchanging the basis vectors f 1 , f 2 of V (0). As a consequence α is mapped to α −1 .
The automorphism s 2 of S is defined by of M at z = 0 one obtains, after conjugation with z 0 0 1 , the local presentation Starting with a local presentation D : This is the matrix of D with respect to the basis zE 2 , E 1 . The induced action of s 2 on R maps α to −α −1 , the formal monodromy at ∞ is multiplied by −1 and the Stokes data are essentially unchanged. The group of automorphisms of S, generated by s 1 , s 2 , has order 4. The Bäcklund transformations are the lifts of the elements of this group to isomorphisms (preserving isomonodromy) between various moduli spacesM(θ).
is the obvious lift of s 1 , given byt →t + πi, θ → −θ. Further any solution q(t) of PIII(D 7 ) for the parameter θ is mapped to the solution q(t+πi) for the parameter −θ.
is the obvious lift of s 2 witht →t + πi. The formula for s + 2 is not obvious and its computation is given below.
The group s + 1 , s + 2 generated by s + 1 , s + 2 (for their action on θ,t) has B ∼ = Z as normal subgroup and s + 1 , s + 2 / B is the affine Weyl group of type A 1 . Computation of the Bäcklund transformation s + 2 . A point ξ ∈ M(θ), lying in the affine open subset defined by c 0 = 0 and c 1 = 0, is represented by the operator in standard form where A is obtained from the above matrix by t → −t and adding Since the two matrix differential operators represent the same irreducible differential module over C(z), there is a T ∈ GL(2, C(z)) = 0, unique up to multiplication by a constant, such that (z d dz Theã,q and the entries of the T * are the unknows in the identity The isomorphism s + 2 respects the foliations. For a leaf one has q = q+2a t and substitution in the first formula producesq = − t(θt+tq −q−t) 2q 2 for this Bäcklund transformation on solutions of PIII(D 7 ).
The Bäcklund transformation s + 2 s + 1 maps a solution q for the parameter θ to the solution , with a = tq −q 2 , with parameter 1 + θ.

Remarks
1. One considers for (M, t) ∈ S the connection (V 0 , ∇) with generic fibre M and the local data )e 2 and for a good choice of e 1 , e 2 one obtains One concludes that the locus of the 2. Algebraic solutions of PIII(D 7 ). One easily finds the algebraic solution(s) q with q 3 = t 2 2 for PIII(D 7 ) with θ = 0. Using the Bäcklund transformations one finds an algebraic solution for PIII(D 7 ) for every θ ∈ Z. According to [5,6], these are all the algebraic solutions of PIII(D 7 ). More precisely, q j (t) = e 2πij/3 e 2t/3 3 √ 2 , j = 0, 1, 2 are algebraic solution for θ = 0. We note that q 1 (t) = q 0 (t + 4πi) and q 2 (t) = q 0 (t + 2πi). Since α = 1 and top 3 = 1, these solutions are mapped to a single point of R(1) corresponding to c 1 c 2 = −3, e = −i and certain values for the invariants 12 , 14 , 23 , 34 (which we cannot make explicit). The isomonodromic family for this solution q is It is not clear what makes this family and the corresponding point of R(1) so special.
3. Special solutions of PIII(D 7 ). Consider an isomonodromy family for which the Stokes matrices are trivial, i.e., c 1 = c 2 = e = 0. Then α = i or α = −i. In the first case one computes that 12 = 14 = 1 2 , 23 = 34 = − 1 2 and one finds a unique point of R(i) and a special solution q(t) of PIII(D 7 ) for θ = 1 2 . Using Bäcklund transformations one obtains a similar special solution for any θ ∈ 1 2 + Z. Y. Ohyama informed us that the condition c 1 = c 2 = 0 implies that the corresponding solution q of PIII(D 7 ) is a univalent function of t and is meromorphic at t = 0. Further θ ∈ 1 2 + Z is equivalent to e = 0. See [2] for details.

Definition of the family
The set S consists of the equivalence classes of pairs (M, t), where M is a differential module M over C(z) and t ∈ C * such that: dim M = 2, Λ 2 M is the trivial module, M has two singularities 0 and ∞, both singularities have Katz invariant 1, the (generalized) eigenvalues are normalized to ± t 2 z −1 at 0 and ± t 2 z at ∞. Further, two pairs (M 1 , t 1 ) and (M 2 , t 2 ) are called equivalent if there exists an isomorphism M 1 → M 2 and t 1 = t 2 .
As in Section 2, we will have to replace T by its universal coveringT = C → T ,t → et. Writẽ S = S × TT . Define for α, β ∈ C * the subset S(α, β) of S consisting of the pairs (M, t) such that

The monodromy space
For (M, t) ∈ S, the monodromy data are given by (compare [10]): the symbolic solutions spaces V (0) and V (∞) at z = 0 and z = ∞ (including formal monodromies and Stokes matrices) and the link L : V (0) → V (∞). We make this more explicit.
The module C((z)) ⊗ M has a basis E 1 , We note that t is used to distinguish between E 1 and E 2 . This basis is unique up to a transformation After fixing θ 0 , the E 1 , E 2 are unique up to multiplication by constants. The symbolic solution space V (0) at Now α = e πiθ 0 is well defined and does not depend on the choices for For the basis e 1 , e 2 of V (0), the formal monodromy and the Stokes matrices are: This product is the topological monodromy top 0 at z = 0.
For the basis f 1 , f 2 of V (∞), the formal monodromy and the Stokes matrices are: This is the topological monodromy top ∞ at z = ∞.
The relations are given by the matrix equality In particular, β = 1 4 α + 2 4 . This defines a variety T , given by the variables α, a 1 , a 2 , 1 , . . . , 4 with the only restrictions 1 4 For fixed values of α, β ∈ C * we obtains a variety T (α, β) defined by the variables a 1 , a 2 , 1 , . . . , 4 and the relations: The group G m × G m acts on T and T (α, β), by base change (e 1 , e 2 , f 1 , f 2 ) → (γe 1 , γ −1 e 2 , δf 1 , δ −1 f 2 ). The categorical quotient of T by G m × G m is R → P with parameter space P = C * × C * given by (α, β). This is a family of affine cubic surfaces R(α, β) (this is the categorical quotient of T (α, β)) given by the equation This image is simply connected. For x 1 x 2 = 0, the fiber is one point. For x 1 x 2 = 0, the fiber is an affine line. It follows that U is simply connected and thus R(α, β) is simply connected, too.
Observations 3.4. 1. The above formula differs slightly from the one given in [10,Section 4.5]. This is due to different choices of the standard matrix differential operator. 2. The transformation t → −t, θ ∞ → −θ ∞ and θ 0 → −θ 0 + 2 leaves the family of matrix differential operators invariant. This has the consequence that a solution q(t) of PIII(D 6 ) with parameters θ 0 and θ ∞ , yields the solution q(−t) of PIII(D 6 ) with parameters −θ 0 + 2 and −θ ∞ . This can also be seen directly from the differential equation.

Verif ication of the formula in Theorem 2.3
On the chart ST 1 of M(θ 0 , θ ∞ ) the matrix differential operator has the form z d dz where o X stands for z d dz (X) and X := d dt (X). On obtains the equations A Maple computation shows that this system of differential equations for q, a −1 is equivalent to The equation follows by substitution. Using the transformation z → z −1 one finds that Q = 1 q satisfies the PIII(D 6 ) equation with θ 0 − 1 and θ ∞ interchanged:

After a remark by Yousuke Ohyama
Let (M, t) ∈ S(α, β) with α = e πiθ 0 , β = e πiθ∞ have the property that M is irreducible. Consider the connection (W, ∇) with generic fiber M and locally represented by The second exterior product of (W, ∇) is trivial and thus Λ 2 W has degree 0. Since M is irreducible one has After multiplying B 1 with a scalar, the matrix of D with respect to the basis B 1 , B 2 has the The result is a new representation of D, namely For unique h 0 , h −1 and at most two values of h −2 , the last operator is Let e 1 , e 2 denote the new basis. Then V = Oe 1 ⊕ O(−[0])e 2 and the corresponding point ξ ∈ M(θ 0 , θ ∞ ) satisfies q(ξ) = 0. The converse holds, too. One observes that q −1 (0) ⊂ M(θ 0 , θ ∞ ) has two connected components, each one isomorphic to A 1 × T . We note that the map Q := 1 q can also be used in this context, since for a monodromic family Q satisfies a PIII(D 6 ) equation (see the end of Section 3.3.3). According to Malgrange, the locus where the bundle W is not free is the tau-divisor. Thus we find that the tau-divisor coincides with the locus Q −1 (∞) ⊂ M(θ 0 , θ ∞ ). The statement: 'the tau-divisor coincides with q −1 (∞)' holds for PI, PII, PIII(D 7 ), PIII(D 8 ), PIV, too (see [8,9,12]).

Geometric quotients of the monodromy data
We use here the notation of Section 3.2.
The remedy consists of replacing T (α, α) by T (α, α) * which is the complement of the closed subset of T (α, α) given by the equations 2 = 3 = a 1 = a 2 = 0. We claim that T (α, α) * has a nonsingular geometric quotient by the action of G m × G m . A proof is obtained by writing T (α, α) * as the union of the four affine open subsets 2 = 0, 3 = 0, a 1 = 0 and a 2 = 0. On each of these subsets one explicitly computes the quotient by G m × G m , which turns out to be nonsingular and geometric. Gluing these four quotients produces the required geometric quotient which will be denoted by R(α, α) * .
Define the closed space T (α, α) * red of T (α, α) * by the condition that the data is reducible. This space has two irreducible components, given in terms of the matrices L, top 0 by: which sends the (L, top 0 ) to ( 2 : a 2 ) ∈ P 1 is the geometric quotient. Therefore the 'reducible locus' R(α, α) * red (i.e., corresponding to reducible monodromy data) is the union of two, not intersecting, projective lines.
2. The case α = β −1 = ±1 can be handled as in (1). One finds (with a similar notation) a geometric quotient R(α, α −1 ) * of T (α, α −1 ) * and a bijectionS(α, The locus of the points in T (1, 1) which describe the monodromy data for modules in S(1, 1) which are direct sums is the union of the two closed sets a 1 = a 2 = 2 = 3 = 0 and a 1 = a 2 = 34 = 0. The group G m × G m acts on each of these open affine sets and the categorical quotient is a geometric quotient and is nonsingular. Therefore the quotient R(1, 1) * of T (1, 1) * , obtained by gluing the six quotients, is a geometric quotient and nonsingular.

Reducible modules in S
We use here the notation of Sections 3.2 and 3.4.1.
Observations 3.5. Let N ⊂ M be a 1-dimensional submodule, then C((z)) ⊗ N = C((z))E i and C((z −1 )) ⊗ N = C((z −1 ))F j with i, j ∈ {1, 2}. Since N has no other singularities than 0, ∞ one has N = C(z)n with δ(n) = ( ±tz −1 ±tz where f ∈ C(z) has no poles at 0 and ∞. Using that N has only singularities at 0 and ∞, one can change the generator n of N such that f is a constant d. Any other base vector of N with this property has the form z k n with k ∈ Z. Further d ∈ ±θ 0 /2 + Z and d ∈ ±θ ∞ /2 + Z and hence α = β ±1 . 2. Let (M, t) ∈ S be reducible, but not a direct sum of two submodules of dimension one. Then there are unique elements 1 , 2 ∈ {−1, 1}, a complex number d, unique modulo Z, and a polynomial c = c 1 z +c 0 = 0, unique up to multiplication by a scalar, such that M is represented by the matrix differential operator 3. For α = ±1, the reducible locus S(α, α) * red of S(α, α) * is represented by the union of the two families in (2) given by e 2πid = α, 1 = 2 = 1 and 1 = 2 = −1. Each of the two families is isomorphic to P 1 × T , by sending the matrix differential operator to ((c 1 : c 0 ), t) ∈ P 1 × T .
Proof . 1. This follows from Observations 3.5 and the statement that a differential module over C(z) is determined by its monodromy data (i.e., ordinary monodromy, Stokes matrices and links) and the formal classification of the singular points (see [10,Theorem 1.7]). 2. For convenience we consider the case i = j = 1 of (1). Using the above Observation, one finds that M has a basis m 1 , m 2 such that z∂(m 2 ) = am 2 and z∂(m 1 ) = −am 1 + f m 2 with a := tz −1 +tz 2 + d and f ∈ C(z). If we fix d, then m 2 is unique up to multiplication by a scalar. Further, m 1 is unique up to a transformation m 1 → λm 1 + hm 2 with λ ∈ C * , h ∈ C(z). This transformation changes f into λf + 2ah + zh .
We start considering the subgroup of transformations with λ = 1 and h ∈ C(z). For a suitable h the term f 1 := f + 2ah + zh has in C * at most poles of order one. A pole of order one of f 1 in C * cannot disappear by a transformation of the form under consideration. Since M has only singularities at 0 and ∞ we conclude that f 1 ∈ C[z, z −1 ]. For suitable h ∈ C[z, z −1 ] the term c := f 1 + 2ah + zh is a polynomial of degree ≤1 and is =0, by assumption. For any h ∈ C(z), h = 0 the term c + 2ah + zh is not a polynomial of degree ≤1. This yields a unique c for this subgroup of transformations. Finally, the transformation m 1 → λm 1 shows that c is unique up to multiplication by a scalar.

Isomonodromy for reducible connections
The fibers of the locally defined map S(α, α ±1 ) * red → R(α, α ±1 ) * red are the isomonodromy families of reducible modules. As a start, we consider the reducible familly of type ( 1 , 2 ) = (1, 1) lying in M(2d + 2, 2d). This family is represented by z d dz + For an isomonodromy subfamily of this, q is a function of t and the Stokes data at 0 and ∞ and the link are fixed. Isomonodromy is equivalent to the statement that the above matrix differential operator commutes with an operator of the form d dt + B −1 z + B 0 + B 1 z, where the tracefree 2 × 2 matrices B −1 , B 0 , B 1 depend on t only. This leads to the equation A computation yields the equation q = −2q 2 − 4d−1 t q − 2. The solutions of this equation have the form 1 2 y y , where y is a non zero solution of the Bessel equation y + 4d−1 t y + 4y = 0. One obtains in a similar way for an isomonodromic family of reducible modules of type ( 1 , 2 ) the equation The solutions are q = 2 2 y y where y is a solution of the Bessel equation y + 4d+1 t y + 4 1 2 y = 0. These equations are consistent with the formula of Theorem 3.3 for isomonodromic families in M(θ 0 , θ ∞ ). According to [5] we found in this way all Riccati solutions for PIII(D 6 ), up to the action of the Bäcklund transformations.
Remark 3.8. The assumption that a function q satisfies two distinct PIII(D 6 ) equations leads to q 4 = 1. Thus we found the algebraic solutions q = ±1 for θ ∞ = θ 0 − 1 and q = ±i for −θ ∞ = θ 0 − 1. According to [5] these are all the algebraic solutions of PIII(D 6 ), up to the action of the Bäcklund transformations.
These elements generate B(S) because: s * is mapped to σ * for * = 1, 2, 3, 4; B 3 = s 2 1 = s 4 3 ; B 1 = s 3 s −1 1 s 4 s 3 s 4 and B 2 = B −1 1 s 2 2 . The group B 3 , generated by B 3 , is isomorphic to Z and lies in the center of B(S). Put B(S) = B(S)/ B 3 and let s * denote the image of s * in this quotient. Then s 2 3 has order two and lies in the center of B(S).
We compare this with Okamoto's paper [7]. The group of the Bäcklund transformations B of equation PIII (D 6 ) is computed to be the affine Weyl group of type B 2 . The substitution x = t 2 , Q = tq transforms the equation into and thus in our notation α = 4θ ∞ , β = −4(θ 0 − 1), γ = 4, δ = −4. In a sense, PIII(D 6 ) is a degree two covering of PIII (D 6 ). It can be seen that there is a surjective homomorphism B(S) → B with kernel s 2 3 . Our approach using moduli spaces explains the Bäcklund transformations presented in [4]. In contrast to this, the new transformations of [13] do not seem to have a simple modular interpretation.

Formulas for the Bäcklund transformations
A point on the first chart of the first space is represented by the differential operator z d dz + az −1 b z − q − az −1 , where a := a −1 , q := −c 0 , b = b 1 z + b 0 + b −1 z −1 + b −2 z −2 and the b 1 , . . . , b −2 are polynomials in t, q, q −1 , using the notation of Section 3.3.1. This is transformed by s 2 into the operator z d dz . We want to compute an operator . . ,b −2 polynomials in t,q,q −1 , representing a point on the first chart of M( 1 2 + θ 0 2 , 1 2 + θ∞ 2 ), which is equivalent to z d dz + A. Thus we have to solve an equation of the type {z d dz + A}T = T {z d dz +Ã} with T ∈ GL(2, C(z)). A local computation shows that T has the form T 0 + T −1 z −1 + T −2 z −2 = 0 with 'constant' matrices T 0 , T −1 , T −2 . A Maple computation yields the solutioñ The induced map for solutions of PIII(D 6 ) is obtained from the formula forq and the equality q = 4a−q t . Comments on the formulas. The term q in the denominator of the formulas is due to our choice of working on the first charts of the spaces M(θ 0 , θ ∞ ) and M(θ 0 + 1, θ ∞ + 1). This term does not produce singularities for s 2 .

s
On the first charts the map is given by: and similarly on the second charts. For the PIII(D 6 ) equations, the map is t → it, A Maple computation shows that there is a unique solution in terms ofq andã of the equation The induced map for solutions of PIII(D 6 ) is obtained from the formula forq and the equality q = 4a−q t . Comments on the formulas. The denominator in these formulas is due to a possible reducible locus of type ( 1 , 2 ) = (−1, −1). This locus is present in M(θ 0 , θ ∞ ) if and only if θ 0 − θ ∞ ∈ 2Z and θ 0 − θ ∞ ≤ 0. This locus is not present in M(θ ∞ , θ 0 ) if moreover θ 0 − θ ∞ < 0. However, in the critical case θ 0 = θ ∞ , s 4 turns out to be the identity.
Using the formulas for s 1 , . . . , s 4 one can deduce formulas for B 1 and B 2 . We will however derive these by the direct method used for s 2 and s 4 . where the entries ofÃ are polynomials inq,q −1 andã =ã −1 . Since the two differential operators represent the same differential module, there exists a T ∈ GL(2, C(z)) such that z d dz +Ã = T −1 (z d dz + A)T . Local calculations at z = 0 and z = ∞ predict that T has the form T = T −2 z −2 +T −1 z −1 +T 0 , T −2 = 0 * 0 0 and det T ∈ C * .
4.4.6 The transformation B 2 : θ 0 → θ 0 , θ ∞ → 2 + θ ∞ ,t →t Let an object of M(θ 0 , θ ∞ ) be represented by a standard operator z d dz + A. We expect that the transformation B 2 yields an object of M(θ 0 , 2 + θ ∞ ), represented by a standard operator z d dz +Ã. Then z d dz +Ã = T −1 (z d dz + A)T for a certain T ∈ GL(2, C(z)). Local calculations at z = 0 and z = ∞ show that T has the form T −1 z −1 + T 0 + T 1 z with The matrix A depends on q and a := a −1 and the matrixÃ depends onq andã :=ã −1 . We have to solve the equation T (z d dz +Ã) = (z d dz + A)T . Maple produced the formulã The denominator ofã is the square of the denominator ofq and the numerator ofã is too large to copy here. The substitution of a = tq +q 4 in the formula forq yields the B 2 map for the solutions of PIII(D 6 ).
Comment. As in Sections 4.4.2, 4.4.4 and 4.4.5, the map B 2 is well defined because B 2 changes the types ( 1 , 2 ) of the reducible loci.