Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

We introduce a sl_2-invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the family constitutes a Lie algebra structure and each vector field from this family is solenoidal, completely integrable and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie algebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this family can be uniquely characterized by two alternative representations, one uses a vector potential while the other uses two functionally independent Clebsch potentials. Our normal form results are designed to preserve these structures and representations. The results are implemented in Maple in order to compute vector potential and the Clebsch potential normal forms of a given vector field from this family. Some practical normal form coefficient formulas for degrees of up to four are presented.


Introduction
We are concerned with nonlinear normal form classification of a sl 2 -Lie algebra generated family of vector fields with a non-semisimple nilpotent linear part, i.e., N := x ∂ ∂y + 2y ∂ ∂z . Jacobson-Morozov theorem provides the other two generators, that they form a triple along with N, for a sl 2 -Lie algebra. Consider N as a differential operator. Then by the adjoint action of the sl 2 -Lie algebra on (nonlinear) vector fields, we introduce a sl 2 -invariant family of vector fields. We show that the set of all such nonlinear vector fields whose linear part is a multiple scalar of N constitutes a Lie algebra, where we denote it by B.
An important goal in our normal form results is to detect, compute and preserve possible symmetries and geometrical features of a vector field's flow. Hence, several geometric properties for our introduced sl 2 -invariant family are carefully studied in this paper. A natural dynamics analysis of such vector fields must respect these geometrical features which have applications in different applied disciplines. However, the classical normal form computations typically destroy certain symmetries of the truncated normal form system. These may include the system's properties such as volume-preserving, Clebsch potentials, vector potentials, etc. Thus in either of these cases, the dynamics analysis of the truncated normal form is not appropriate. Hence it is important to use permissible transformations which preserve the system's symmetry; also see [14,16,19]. One of our most important claimed contributions here is that our (truncated) normal form results preserve all the different representations (described below) in the paper such as vector potential, Clebsch potentials, and the volume-preserving property.
Solenoidal vector fields appear in disciplines such as magnetic fields and fluid mechanics; e.g., see [21,[23][24][25]. Computing the invariants of such vector fields is an important goal in this paper. Any solenoidal vector field, say v, takes a vector potential representation, that is, there exists a vector field, say w, whose curl is v, i.e., ∇ × w = v. We prove that all vector fields in B are solenoidal and provide the method and formulas for deriving their vector potential normal forms. We further introduce a Poisson algebra that is Lie-isomorphic to B through a Lie isomorphism ψ. Another most important claimed contributions in this paper is that the Lie isomorphism ψ associates a first integral ψ(v) to each vector field v from B. We further show that the quadratic polynomial ∆ := xz − y 2 is a second first integral for all vector fields in B.
Analytic normalization of an analytic vector field has close relations with complete integrability of the vector field; e.g., see [34,35,42,46]. We recall that two first integrals for v in B are called functionally independent when their gradients have a rank of 2 for almost everywhere; e.g., see [42, page 3553] and [34]. The level curves of these invariants provide a comprehensive understanding about the orbits in the state space associated with the vector field's flow. Each vector field v from the sl 2 -invariant Lie algebra B is a completely integrable solenoidal vector field; i.e., we show that the invariants ∆ and ψ(v) for each v ∈ B are functionally independent. There is another alternative representation for completely integrable solenoidal vector fields, that is given by the two of the vector field's functionally independent invariants. Indeed, we prove that each vector field v in B equals the exterior product of the gradients of ∆ = xz − y 2 and ψ(v); the latter is obtained through the Lie isomorphism ψ between B and our introduced Poisson algebra. The first integrals in the exterior product are referred as Clebsch potentials or Euler potentials of the vector field v; e.g., see [23,24]. We refer to ∆ by the primary Clebsch potential and ψ(v) as the secondary Clebsch potential for v. We further conclude that these families of triple zero singularities are rotational vector field, that is, their curl is non-zero. This implies that these are not gradient vector fields.
Finally, we prove that B is the set of all multiple scalars of solenoidal vector fields such as v, that is given by where v : R 3 → R 3 denotes a vector field without constant and linear parts, div(v) = 0, and v(∆) = 0 where ∆ = xz − y 2 . (1.2) Note that for our convenience we interchange the uses of notations and terminologies such as vector fields, differential systems and differential operators. We also remark that x and ∆ are the two generators of the invariant algebra for the linear vector field N. We refer to the vector field v in equation (1.1)-(1.2) by a completely integrable system since it has two functionally independent invariants ∆ and ψ(v). The Poisson structure and the Lie isomorphism ψ provide a practical method for deriving the second first integral within the invariant algebra of vector fields given by (1.1)-(1.2) and their normal forms. Normal form classification of nilpotent singularities has been a challenging task. Even in the twodimensional case, there have been numerous important contributions in various types and approaches; e.g., see [1-3, 5, 10, 13, 18, 20, 22, 36-39, 41, 43, 45]. There have only been a few contributions in three dimensional state space cases; see [11,44] where hypernormalization is performed up to degree three; also see [15][16][17] and [27][28][29][30][31]. In this paper we provide a complete normal form classification for all vector fields v in equations (1.1)-(1.2), that is, the set of all completely integrable solenoidal nilpotent singularities where ∆ is one of their invariants and a multiple scalar of N is their linear part. These vector fields and their normal forms are uniquely characterized by their secondary Clebsch potential. Indeed, the primary Clebsch potential ∆ is always preserved throughout the normalization steps while the normalizing transformations naturally reflect the normal form changes into the secondary Clebsch potential. In Theorem 5.1, we prove that a vector field given by (1.1)-(1.2) can be either linearized or uniquely transformed into the formal normal form vector field where b i,k ∈ R and p is a natural number. In addition, the secondary Clebsch potential normal form is given by The normal form invariant (1.4) can sometimes be used for further reduction of the normal form vector fields; e.g., see section 5. Now we describe the organization of the rest of this paper. We introduce a family of sl 2 -invariant irreducible vector spaces of vector fields in section 2. We further prove that this family constitutes a Lie algebra and derive their associated structure constants. Section 3 is devoted for introducing a Poisson algebra and prove that it is Lie isomorphic to B. Next, we discuss the geometrical properties of B family in section 4. In particular, we show that our sl 2 -invariant introduced family of vector fields are fully characterized by equations (1.1)-(1.2). Two further representations for each such vector field are presented in this section by using their Clebsch potentials and vector potentials. Section 5 is dedicated to study the normal form classification for vector fields (1.1)-(1.2). Some practical formulas for normal form coefficients of up to degree three for a given triple zero singularity (1.1)-(1.2) is presented. Finally, we introduce two more sl 2 -invariant families of vector fields in Appendix A. These along with the family given by equations (1.1)-(1.2) would amount to three family types of sl 2 -invariant vector fields so that each three dimensional vector field can be uniquely Taylor expanded in terms of vector polynomial generators from these three types. We denote N n f v = (N n f ) v for the iterative action of N as a differential operator on the scalar function f that is also multiplied with v. Further for a vector field v, N n v is inductively defined by Nv := ad N v, and N n v := ad N N n−1 v for n > 1.

Algebraic structures
We often put the underline only on the last element on which the operator acts, i.e., N n f gh := hN n f g and Nf M := Nf M. Note that N as an operator distinguishes vector fields from scalar functions: the operator N merely acts on scalar functions as a differential operator while it acts as a Lie operator on vector fields. By [ where ω f and f are called the eigenvalue and eigenfunction of the differential operator H, respectively. The algebra of first integrals for M is the same as ker M = ∆, z ; see [7], [ Notation 2.1.
• The following notations frequently appear in this paper.
• We denote e 1 , e 2 and e 3 for the standard basis of R 3 and κ l,i := i! (i−l)! .
• We use the Pochhammer k-symbol notation for any a, b ∈ R, k ∈ N as (a) k b := k−1 j=0 (a + jb). • Throughout this paper we frequently use some constants or variables with negative powers in the denominator (or numerator) of a fraction. The reader should merely treat this as a formal misuse of notation to shorten the formulas. Now we present some technical results which play a central role in this paper. There is an important corollary to this lemma.

Corollary 2.3.
For any homogeneous function f ∈ ker M, we have Lemma 2.4. For each l ∈ N 0 , the following equalities hold: Proof. The proof is by induction on l. For instance, by the induction hypothesis we have This proves the second equality.
Lemma 2.5. Let q = 2s + r, where r = 0 or r = 1 and s ∈ N 0 . Then, for i ≥ s + n + r, while η q,i n := ζ q,i n := 0 for i < q + n − s. In particular, N q z i = 0 when s < q − i.
Proof. The proof is straightforward by an induction on q.
Lemma 2.5 enables us to formulate N q 1 z i N q 2 z j in terms of x, y, z and ∆.
Proof. Given Lemma 2.5, the proof is trivial.
The following theorem provides an alternative formula for the expansion of N q 1 z i N q 2 z j .
Theorem 2.7. Let q 1 = 2s 1 + r 1 , q 2 = 2s 2 + r 2 and σ 2 (q 1 , q 2 , i, j) = q 1 + q 2 − i − j. Then, where C q 1 ,q 2 p,i,j is given by Proof. A polynomial expansion for the left hand side in (2.9) is derived in equation (2.7) while by using the first equation in (2.5), the right hand side is given by When r 1 r 2 = 0, equation (2.9) is equivalent with the following polynomial equation x s 1 +s 2 −n z i+j−(s 1 +s 2 +r 1 +r 2 +n) ∆ n .
Hence for each i, j, s 1 , s 2 , and 0 ≤ n ≤ min{s 1 + s 2 − σ 2 , s 1 + s 2 }, we havẽ These introduce a family of upper triangular linear matrix equations. The determinant of the coefficient matrix is given by min{s 1 +s 2 −σ 2 ,s 1 +s 2 } n=0 η q 1 +q 2 −2n,i+j−2n 0 = 0. These together with the first equation in (2.6) give rise to Hence the family of linear equations and its solutions are derived bỹ respectively. .
Here, z has an eigenvalue 2 while the eigenvalue for ∆ is 0. Hence, differentiating with respect to u at u = 1 leads to 3 . The latter is the generating function for all formal power series associated with three variables. The proof is complete by a differential form version of [28,Lemma 4.7.9].
where N 0 denotes nonnegative integers. Then, K = ker ad M and V is the set of all three dimensional formal vector fields.
Proof. We first claim that the homogeneous polynomial set Then, the proof of our claim is complete by the linear independency of polynomials N n z i+1 , 2yN n z i and xN n z i . Since M commutes with M, E, ∂ ∂x , and Mz i = 0, K is a subspace for ker ad M . By equation (2.11), the generating function for R[[z, ∆]] is 1 (1−u 2 t)(1−t 2 ) and thus, the generating function for ker ad M is given by . This concludes that K = ker ad M . Furthermore, V is the space of all three dimensional vector fields with three variables and its generating function is given by 1 ( (2.14) Note that two more families of vector fields associated with Theorem 2.9 are defined in Appendix A.
Hence, the constant a q 1 ,q 2 −1,i 1 ,i 2 are derived through equation (2.18). Now the proof is complete by derivation of the formula for the first component of , and the equality  Hence, the structure constants associated with monomials are given by for arbitrary nonnegative integers m, n, p, i, j, k. Now define where ad Proof. Due to the previous lemma, the actions of ad x and ad N on z i are identical. Hence, our claim readily follows from Lemma 2.5.

Now we define a vector space B as
The following two lemmas show that B is a Poisson algebra and it is Lie-isomorphic to B.

Lemma 3.2. The space B is invariant under the Poisson bracket and the linear map
is a Lie isomorphism.
Proof. By the Leibniz rule we have ad n x z i+1 = z ad n x z i + n ad x z ad n−1 (x)z i + n(n − 1) 2! ad 2 x z ad n−2 x z i = z ad n x z i + 2ny ad n−1 x z i + n(n − 1)x ad n−2 x z i . Due to equation (3.7), we have Ψ(x) = N, Ψ(y) = H 2 , and Ψ(z) = −M. Further, ad n x z i+1 ∆ k = ad n x z i+1 ∆ k . The actions of ad n x on z i+1 ∆ k and ad N on z i ∆ k M are identified through Ψ. Then, the proof follows an induction on n, structure constants (3.1) and those governing the sl 2 triple M, N, and H. Now we present a ring structure constants for B so that B is a Poisson algebra. Lemma 3.3. The space B is a Poisson algebra. In particular, let q 1 = 2s 1 + r 1 and q 2 = 2s 2 + r 2 . Then, the ring structure constants are given by ⌋ . Proof. The proof directly follows from (3.4) and the formulas given in Theorem 2.7. Indeed, we have The following theorem presents a property that is similar to the Hamiltonian cases, i.e., the rate change of functions along with vector fields from B can be computed by the Poisson bracket.  A vector field v is called solenoidal (nondissipative, incompressible, or volume-preserving) when div(v(x)) = 0, and otherwise the vector field v is called generalized dissipative, i.e., div(v(x)) = 0. When v(x) = ∇f (x) for a scalar function f (x), the vector field v is called a gradient or a globally potential vector field. Examples of this are the gravitational potential, a mechanical potential energy, and the electric potential energy. The vector field v(x) is said to be nonpotential (non-gradient) when there exists at least a point x ∈ R 3 such that curl(v(x)) = 0; e.g., see [40, page 1].
Proof. By the Leibniz rule and B l i,k = ∆ k B l i,0 , we observe that We claim that ∇ · B l i,0 = 0 and ∇(∆ k ) · B l i,0 = 0. Using Lemma 2.4, the partial derivatives of N l z i+1 are given by

Equation (2.13) and Lemma 2.4 give rise to
On the other hand for l = 2s, equation (2.13) gives rise to The last equality is derived from Lemma 2.5. Hence, ∇(∆ k ) · B l i,0 = 0 due to When l = 2s + 1, ∇(∆ k ) · B l i,0 is given by Hence ∇(∆ k ) · B l i,0 = 0 due to the equation  Indeed for every v ∈ B, Ψ −1 (v) ∈ B and ∆ are two first integrals for v.
• A Clebsch potential representation for B l i,k is given by Equation (4.6) provides an alternative representation for each vector field v in B by using the primary and secondary Clebsch potentials ∆ and Ψ −1 (v) ∈ B.
• The polynomial functions b l i,0 and ∆ are two functionally independent first integrals for B l i,0 .
When i, k ∈ N 0 , −1 l 2i + 1, and N := i + 2k, we define a condition for a nonnegative integer m by and next, a set P l i,k by
Proof. The claim in part 1 directly follows from equation (2.13). For claim 2, let B l ′ i ′ ,k ′ be the vector field in B with (l, i, k) = (l ′ , i ′ , k ′ ). Here we only consider the case p = 3. Using the polynomial expansion for ∆ k , ∆ k ′ , Lemma 2.5 and definition (2.12), the monomials appearing in B l i,k and B l ′ i ′ ,k ′ follow for some n 1 , n 2 , p 1 , p 2 , where l = 2s + r and l ′ = 2s ′ + r ′ . Let P = Q. Then, By substituting the first equation in (4.14) into the third one, we have Since the vector fields B l i,k and B l ′ i ′ ,k ′ have the same δ-grade, Therefore, i−i ′ = l −l ′ = 2(k ′ −k) and i+2k = i ′ +2k ′ . Hence, the inequalities i ′ ≥ 0 and −1 ≤ l ′ ≤ 2i ′ +1 are equivalent to the inequalities (4.12) on m := k ′ . The equality P l i,k = P m 1 m 2 ,m 3 is due to the definition. Part 3 follows equations (4.16) and (4.15). The claim 4 is a direct corollary of the claim 2. The proof of part 5 is similar to claims 2-4 and the claim is consistent with equation (4.6).
Finally for the claim 6, consider v := ∞ N =0 w N for w N ∈ B N . Then by claim 5, for any N we have Now we Taylor-expand w j N in terms of ∆, that is, Hence, v j N −2k,0 does not share any monomial vector field with any B-terms except B j N −2k,0 . This is because of the claim in part 2 and that we have already excluded the powers of ∆ in (4.17). Therefore, the expansion of v j N −2k,0 in terms of the B-term generators of B only includes B j N −2k,0 .
Remark 4.5. Given Lemma 4.4, the same statements trivially hold for other sl 2 -generated vector fields from spaces A and C defined in Appendix A. In particular, for any v ∈ C and w ∈ A , there exist uniquely determined constants c j N,k and a j N,k so that The following theorem provides a concrete characterization for vector fields in B.
Hence, ∇ · (w C + w A ) = 0 and w C (∆) + w A (∆). By equations (4.18) and (4.19),  Proof. The first item follows items 3 and 4 in Theorem 4.4, and Lemma 3.2. The second and third claim follows from the structure constants and Lemma 3.2.
An alternative representation for vector fields in B are based on the vector potential. Each solenoidal vector field v has always a vector potential that is unique modulo gradient vector fields. Vector potential frequently appears in the classical and quantum mechanics, e.g., see [25]. Vector potential is called magnetic vector potential in electrodynamics while the curl of the magnetic vector potential is called magnetic field; see [4].
Proof. From equation (4.6) and the equality ∇f × ∇g = ∇ × f ∇g for all scalar functions f and g, we have Remark 4.9. An alternative vector potential for solenoidal vector fields is available through the computational approach on [26, page 21]. Indeed, there exists a vector potential Φ l i,k such that B l i,k = curl(Φ l i,k ), where In particular, Φ 1 0,0 = (−∆, 0, 0). The proof here follows [26, page 21]. Indeed, define where X := (x, y, z). Then, a vector potential for B l i,k is given by P (X) × X. Hence, equation (4.21) is computed through equation (2.13).
We further recall that a vector potential for a given solenoidal vector field is generally unique modulo gradient vector fields. For instance by equations (4.20) and (4.21), vector fields and curl(v j N −2k,0 ∆ k ) = 0 for all j, l, N. Now let curl(B l i,k ) = 0. Hence, all three components of curl(B l i,k ) are zero. The first component of curl( (i+1)κ l+1,2i+2 and by Lemma 2.4, we have Since Terms(∆ k N l−1 z i ), Terms(∆ k−1 N l z i+1 ), Terms(∆ k N l+1 z i ), and Terms(N l+1 z i+1 ) are pairwise disjoint sets of monomial terms, equation (4.22) holds if and only if l(l + 1) = k(l + 1) = (i − l) = k(i − l) = 0. The later is equivalent with i = k = l = 0. This completes the proof.

Consider the vector field
This family has two first integrals of y and z while A −2 i,0 is also solenoidal for all i. These vector fields do not generate a Lie algebra with the nilpotent linear part B 1 0,0 , indeed, This indicates that the family of vector fields in B does not represent the set of all solenoidal vector fields with two independent first integrals.

Normal form classification
where b i,k ∈ R. Furthermore, the normal form vector field (5.1) is always an the infinite level normal form. In addition, the infinite level secondary Clebsch potential normal form is given by Here, the quadratic polynomial ∆ = xz − y 2 stands for the primary Clebsch potential.
Proof. The normal form is readily available given the sl 2 -style normal form and the fact that ker(ad M ) = span{B −1 i,k }; for more information see [2,3]. Let r := min{i | b i,0 = 0} and r > 0. Define a new grading function by δ(B l i,k ) := lr + 2i + k. Then, B is a δ-graded Lie algebra and Linear invertible transformations can be used to rescale the coefficient b p,0 into b p,0 := 1. Following [2,3] we define By the structure constants and equation (2.19), Hence for any i and k (q = 2i + 1), there is a possibility of a vector polynomial in kernel Γ. This is due to a similar argument used by [2,3]. On the other hand b 2i+1 ∈ ker Γ. These polynomial vectors are extended to a symmetry for the normal form vector field (5.1), through This proves that there is no possibility of any hypernormalization beyond the sl 2 -style normalized vector field (5.1).
Corollary 5.2. The following presents five alternative representations for the normal form (5.1): 1. A formal sum of B-terms: 2. The secondary invariant: Here, Ψ is the Lie isomorphism given by equation (3.7).
Theorem A.2. Each three dimensional vector field v can be uniquely expanded in terms of formal sums of polynomial generators A l i,k , B l i,k and C l i,k from A , B and C , respectively.