Differential Galois Theory and Lie Symmetries

We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the differential Galois group in the system in a way that depends of the Maclaurin series of the symmetry along the zero solution.


Introduction
Differential Galois theory and Lie symmetries are two different theoretical frameworks designed to deal with similar mathematical problems: the integration, reduction, classification and listing of solutions of differential equations. Both theories appear simultaneously at the end of 19th century, but the links between them remain hidden, mostly because of the apparent walls that separate mathematical disciplines. Differential Galois theory appears to be central to differential algebra. On the other hand, the theory of Lie symmetries belongs to the realm of local differential geometry. For a general exposition of both theories we refer the readers to [6,12] and [7,11] respectively. Along this paper for differential Galois theory we understand the Galois theory of systems of linear differential equations, also called the Picard-Vessiot Theory.
The interplay between Lie symmetries and Galois group is implicitly suggested by the Galoisian obstructions to the existence of first integrals that for Hamiltonian systems that, by the duality given by the symplectic structure, it translates into obstructions for the existence of suitable Hamiltonian vector fields of Lie symmetries. About this approach to non-integrability of dynamical systems, see the recent survey [10], references therein. See also the paper [2] which gives a more general definition of integrability by including directly some Lie symmetries. In this paper we shall not explore this interesting connection.
The problem of linking differential Galois theory with Lie symmetries of differential operators has been studied by C. Athorne in [1] and, by W. R. Oudshoorn and M. van der Put in [13]. There is also link between Lie symmetries and the non-linear differential Galois theory proposed by B. Malgrange (see for example remark iii page 224 in [9] for a comparison between Lie symmetries and Galois symmetries). The general idea is that differential Galois groups and Lie symmetries should be related in the sense that the more symmetries are present the smallest the Galois group should be. Our results are an elaboration of this general idea into precise statements that apply to the particular case of systems of linear differential equations.
In [1,13] the authors looked for the link between the symmetries of a higher order scalar linear differential equation and its Galois group. Here, we will take a slightly different point of view. Instead of considering higher order linear differential equations, we will consider systems of first order linear differential equations. This in turn will allow us to clarify the interplay between Lie symmetries and the differential Galois group.
There is a reason to consider systems instead of higher order equations. Although the differential Galois theories of systems or higher order equations are equivalent, their Lie symmetry theories are not. Higher order equations have much less symmetries than systems of first order differential equations. There are still many interesting questions about the interplay of those two approaches.
In order to link Lie symmetries with differential Galois theory we take a geometrical approach to the latter which is developed by the first two authors in [4]. Some general results about symmetries were already stated in [5] Section 6, but in a more general context of automorphic systems. In particular, it is implicit in the said paper that the eigenring consists of vertical Lie symmetries. Our approach here is less abstract and more explicit (see subsection 6.2 later). In connection with that we point out that the relevance of the eigenring for the symmetries of linear differential equations was discovered independently by C. Jensen in the nice paper [8], but without any mention to the relationship with Picard-Vessiot theory.
The results contained in this work may be summarized as follows: (a) The search for symmetries of systems of linear differential equations may be reduced to the search for a particular kind of symmetries, namely homogeneous vertically polynomial symmetries (Lemma 3.1, Proposition 3.1).
(b) The differential equations determining homogeneous vertically polynomial symmetries are again systems of linear differential equations. The differential Galois theories of those equations is related to that of our original system (Theorem 6.1).
(c) The Galois group determines the Lie algebra of rational symmetries (Theorem 6.2).
(d) Each non-trivial symmetry gives a restriction to the Galois group. Thus, the bigger the symmetry algebra, the smaller the Galois group. In many cases a single symmetry forces the group to be abelian or solvable (Theorems 6.3, 6.4, Corollaries 6.1, 6.2) Section two contains the basic definitions of symmetries. Section 3 studies polynomial symmetries and establishes part (a) above. Section 4 gives a dictionary between graded polynomial vector fields, linear actions and corresponding linear differential systems. A geometric definition of the differential Galois is given in section 5 and the comparison with symmetries are derived in section 6.

Characteristic and vertical symmetries
Let U be an open subset of the complex projective line CP 1 . By a function field K we mean a subfield of the field of meromorphic functions in U such that K contains the constants C and it is closed with respect the derivation d dx . Clearly, fields of rational functions, elliptic functions, etc. are function fields. Let us fix from now on the function field K and its domain U . We consider a system of linear differential equations with coefficients in a function field K: = A(x)y, y = (y 1 , . . . , y n ), A(x) ∈ gl(n, K).
The poles of A(x) are called the singularities. The point at infinity is considered as a singularity depending of its behaviour after a suitable change of coordinates. The graphs of the solutions of (1) are the integral curves of the associated vector field: which is a meromorphic vector field in U × C n . An infinitesimal symmetry of X is an analytic vector field Y defined some open subset of U × C n such that [Y, X] and X are linearly dependent in their common domain of definition. In particular, vector fields of the form γX where γ is an analytic function defined in some open subset of U × C n are infinitesimal symmetries of X. They are called characteristic symmetries of X. Since the definition of infinitesimal symmetries is local, we have sheaves of infinitesimal symmetries and characteristic symmetries of X in U × C n .
The Lie bracket of two infinitesimal symmetries is also an infinitesimal symmetry. Hence, infinitesimal symmetries form a Lie algebra sheaf. Characteristic symmetries form an ideal of the Lie algebra sheaf of infinitesimal symmetries. An infinitesimal symmetry Y is a vertical symmetry if it is tangent to the fibers of the canonical projection U × C n → U , that is Y x = 0. Its expression in coordinates takes the form: If Y is a vertical symmetry, then the Lie bracket [X, Y ] vanishes. The Lie algebra sheaf of vertical symmetries is canonically isomorphic to the quotient Lie algebra sheaf of all infinitesimal symmetries modulo the ideal of characteristic symmetries. If Y is an infinitesimal symmetry, we can take its vertical representative: By this reduction the algebra of vertical symmetries is isomorphic to the algebra of infinitesimal symmetries modulo the ideal of characteristic symmetries. Thus, in order to study the symmetries of (1) it suffices to consider vertical symmetries. Also, in order to study vertical symmetries of linear differential equations, we only need to consider the sheaf structure with respect to U . Hence, we introduce the following notation.
Definition 2.1. The sheaf sym X in U assigns to each open subset V ⊆ U the Lie algebra of vertical infinitesimal symmetries of X defined in V × C n , where X(V × C n ) stands for the Lie algebra of analytic vector fields in V × C n .
Our objective is to describe the sections of this sheaf sym X and its relation with the closed form solutions and the Picard-Vessiot theory of the system (1). From now on, when we mention a symmetry of X we mean a section of sym X , that is, a vertical infinitesimal symmetry.
We say that Y is a vertically polynomial vector field when the components f i (x, y) are polynomials in the variables y 1 ,. . ., y n .
Example 3.1. The homogeneous vector field: is vertically polynomial, indeed vertically linear. It is also is a symmetry of any system of linear differential equations. Hence, it is a global section of sym X .
The definitions of degree and homogeneous components of a vertically polynomial vector field are clear. Given a function field F of meromorphic functions in V ⊆ U , we can also speak of the vertically polynomial vector fields with coefficients in F. They are the vertically polynomial vector fields Reciprocally, given a vertically polynomial vector field Y in V × C n with V ⊂ U , there is a smaller differential field extension K{Y } of K that contains all the coefficients of Y .

Homogeneous components of symmetries
Let us consider a vertical infinitesimal symmetry Y ∈ sym X (V) with V ⊆ U . We can develop the Maclaurin series for the components of Y with respect the variables y 1 , . . . , y n , obtaining: where the functions g j,α (x) are analytic functions in V. We join the terms, writing Y as a sum of its homogeneous components, where each Y j is an homogeneous vertically polynomial vector field of degree j (with respect to the y variables) in V × C n . Since the Lie bracket can be computed componentwise, we have: For each j ≥ 0, [X, Y j ] is a homogeneous vertically polynomial vector field of degree j. Thus, all the terms of the above series vanish and we have proved the following result.
All the terms Y j of its Maclaurin series are symmetries of X.
whose polar set does not contain the curve CP 1 × {0} in CP 1 × C n . then the homogeneous components, are vertically polynomial symmetries with coefficients in C(x).

Homogeneous vertically polynomial symmetries
The sheaf sym X contains the subsheaf of vertically polynomial symmetries that we denote by sym <∞ X . Lemma 3.1 implies that the homogeneous components of vertically polynomial symmetries are also symmetries. Hence, we have a decomposition: where sym r X stands for the sheaf of homogeneous vertically polynomial symmetries of X of degree r. These objects can be interpreted simultaneously in two complementary ways, as sheaves and as differential algebraic varieties: (a) As a sheaf, sym r X maps each open subset V ⊆ U to the set sym r X (V) of homogeneous vertically polynomial symmetries of X defined in V × C n .
(b) As a differential variety, sym r X maps each differential field extension K ⊆ F to the set sym r X (F) of homogeneous vertically polynomial symmetries of X with coefficients in F. This makes perfect sense even if F is not a function field.
If the function field K contains the rational functions, then rational symmetries, as vector fields in CP 1 × C n , can be always reduced to vertically polynomial symmetries with coefficients in K.
Proposition 3.1. Assume that K contains the field of rational functions C(x), and let Y be a rational vector field in CP 1 × C n which is non-characteristic rational symmetry of X whose polar subset does not include the curve U × {0} in U × C n . Let us consider the Maclaurin series: where each Y r is a homogeneous vertically polynomial vector field of degree r. Then, for each r, Y r ∈ sym r X (K), and for at least one index r, Y r is not zero.
Proof. Lemma 3.1 and its Remark 3.1 show that the vector fiels Y r are symmetries. We need only to check that they have coefficients in K. Let us consider the expression of Y in coordinates, A direct examination of the expressions shows that they have coefficients in K.
Since the Lie bracket is a graded operation, the sheaves sym r X are not in general Lie algebra sheaves. We have: For n > 1 only sym 0 X , sym 1 X and sym 0 X ⊕ sym 1 X are Lie algebra sheaves. Our next objective is to show that the sections of sym r X for each r are solutions of a hierarchy of linear differential systems canonically attached to (1).

The Lie algebra of polynomial vector fields
Vertically polynomial symmetries are polynomial vector fields along the fibers of the projection U × C n → U . In this section we will give some remarks about the structure of the Lie algebra X[C n ] of polynomial vector fields in C n : By taking homogeneous components, where, The Lie bracket respects the degree in the following way: Remark 4.1. For n > 2, exactly two of the homogeneous components of X[C n ] are Lie subalgebras: (a) The homogeneous component of degree zero X 0 [C n ]. It is the Lie algebra of the infinitesimal generators of the action of the group of translations in C n . It is an abelian Lie algebra canonically isomorphic to C n , where {e 1 , . . . , e n } stands for the canonical basis of C n .
(b) The homogeneous component of degree one, X 1 [C n ]. It is the Lie algebra of linear vector fields in C n . It consists of the infinitesimal generators of the action of the group of linear transformations of C n . It is canonically isomorphic to gl(n, C) in the following sense: where A stands for n × n matrix of entries a ij . Given an endomorphism A ∈ gl(n, C), we let v A denote its corresponding linear vector field in C n : We may easily check that this morphism is in fact an anti-isomorphism of Lie algebras, for any pair A, B of matrices we have:

Induced linear actions
Let us consider the canonical action of GL(n, C) on C n by linear transformations.
GL(n, C) × C n → C n , (A, y) → Ay, By abuse of notation, we denote non degenerated matrices and their associated linear transformation on C n by the same symbols. Let A be a linear transformation and Y be a homogeneous polynomial vector field of degree r.
Viewing A as a (linear) diffeomorphism of C n , we let A * (Y ) denote the transformed vector field by A; in general, for any diffeomorphism F , we let F * (Y )(F (p)) = dF (Y (p)), i.e. F * (Y )(p) = dF (Y (F −1 (p)). This defines a natural action of diffeomorphisms of C n on vector fields of C n . It is easy to check that A * (Y ) is also a homogeneous polynomial vector field of degree r. Thus, for each r ≥ 0 we have an induced representation which yields a linear representations of GL(n, C) on the finite dimensional vector spaces X r [C n ]. This action can be differentiated at the identity obtaining the infinitesimal action, The following remark is key to connect the definition of symmetry with the differential Galois theoretic aspects of of equation (1).
Lemma 4.1. The infinitesimal action Φ ′ of gl(n, C) in X r [C n ] coincides up to a change of sign, by the canonical isomorphism between X 1 [C n ] and gl(n, C), with the action of linear vector fields in X r [C n ] by the Lie bracket. That is, for any endomorphism A and homogeneous polynomial vector field Y in C n , Proof. Let us define, for each A and ε, σ ε : C n → C n , the map that sends each y ∈ C n to e εA y. Thus, {σ ε } ε∈C is the flow of the vector field v A . By definition of α ′ r we have: by the usual geometric definition of Lie derivative. By means of this morphism, we transport the differential equation (1) to a linear differential equation in U × E with coefficients in K: We say that system (4) is the Lie-Vessiot system induced by (1) in the representation (E, Ψ). They are the geometric analog of the differential systems obtained by Tannakian correspondence on tensor constructions in standard differential Galois theory, see [12,6].
A solution of (4) in V ⊂ U is an analytic map V → E that satisfies the equations. Given a differential field extension K ⊆ F, a solution of (4) in F can be thought as an element of E ⊗ C F. If we take a basis v 1 , . . . , v n of E and denote by λ 1 , . . . , λ n their corresponding linear coordinates, the functions b ij (x) = λ i (Ψ ′ (A(x))v j ) are elements of K and the differential equation can be written in coordinates, The non-apparent singularities of (5) form a subset of the singularities of (1). There is a natural relations between their solutions. If M (x) is a fundamental matrix of solutions of (1) defined in V ⊂ U then for all v 0 ∈ E, v(x) = Ψ(M (x))v 0 is a particular solution of (4).

The Galois group
The Lie-Vessiot hierarchy of systems induced by (1) form a hierarchy of differential equations that summarize the differential algebraic properties of (1). It allows us to give a "geometric" definition of the differential Galois group. In this definition we are concerned with two kind of solutions of the Lie-Vessiot systems: (a) We say that a solution v(x) of (4) is K-rational if it belongs to E ⊗ C K. It means, that v(x) has its coordinates (5) in K.

(b) We say that an element
It models the case in which v(x) = exp( b(x)dx)w(x) is a solution of (4), or equivalently, the class v(x) is a rational solution of the equation in the projective space P(E) obtained by reducing (4) by the homogeneous vector field (see Example 3.1) of symmetries.
Note that the concept of K-exponential pre-solution extends that of K-rational solution. Any K-rational solution is a K-exponential pre-solution in which the multiplier b(x) vanish.
Definition 5.1. Let us fix x 0 ∈ U which is not a singularity of (1). We say that a nondegenerated matrix σ ∈ GL(n, C) is Galoisian at x 0 if for any linear representation (E, Ψ) it satisfies the two following conditions: (a) For any K-rational solution v(x) of its induced Lie-Vessiot system (4), Ψ(σ)(v(x 0 )) = v(x 0 ). Note that, if v(x) is a K-rational solution then v(x 0 ) is well defined as an element of E.
(b) For any K-exponential pre-solution w(x) of its induced Lie-Vessiot system (4), for which w(x 0 ) is well defined, w(x 0 ) is an eigenvector of Ψ(σ).
The set of Galoisian matrices at x 0 form a group Gal(x 0 , X), called the Galois group of (1) at the point x 0 . It is the stabilizer of all the values at x 0 of K-rational solutions, and the lines spanned by the values at x 0 of K-exponential pre-solutions of induced Lie-Vessiot systems. Although this geometric definition may seem different from the standard ones from Picard-Vessiot theory, it produces the same group. Choose a normalized local solution matrix at x 0 , i.e. one with initial condition being the identity at the point x 0 ; then v(x 0 ) will be the coordinates of the invariant v(x) on this normalized basis of solutions. Our definition hence says that a matrix σ is in Gal(x 0 , X) if and only if it admits all the invariants of the (Picard-Vessiot) differential Galois group as invariants.
The following facts are well known in differential Galois theory (again we refer the interested reader to [12,6] for a general exposition, or to [4,5] for an exposition which is consistent with our geometric definition): (a) The Galois group Gal(x 0 , X) is an algebraic subgroup of GL(n, C).
(b) The Galois groups at two different non-singular points x 0 and x 1 are conjugated.
We will write Gal(X) to denote this abstract Galois group, that we call the Galois group of the equation (1) with coefficients in K.
(c) The system (1) is integrable by Liouvillian functions if and only if Gal(X) is a virtually solvable group.
The following result will by important in our further considerations. The reader may check [4] Proposition 5.4 for a geometric proof that relies on Lie's reduction method and Chevalley theorem.

Symmetries as solutions of the Lie-Vessiot hierarchy
The key of the relation between the Galois group and the symmetries, is that vertically polynomial vector fields in V ⊆ U of can be seen as maps V → X[C n ]. For a given vertically polynomial vector field Y and x 0 ∈ V we will write Y (x 0 ) for the value of Y at x 0 . It is a polynomial vector field in C n that corresponds to the restriction of Y to the fibre {x 0 } × C n . It is clear that, for general vertical vector fields, If we restrict our considerations of homogeneous vertically polynomial vector fields of a fixed degree r then X r [C n ] turns out to be a finite dimensional complex space. This will allow us to describe vertically polynomial symmetries as solutions of some systems of the Lie-Vessiot hierarchy. Theorem 6.1. Let Y be a homogeneous vertically polynomial vector field of degree r defined in V × C n with V ⊆ U . Then, Y is a symmetry of (1) if and only if, as a map from V to X r [C n ], it is a solution of the Lie-Vessiot system induced in the representation (Φ r , X r [C n ]). That is, Y ∈ sym X (V) if and only if it satisfies: Proof. Let us call v A (x) the vertically linear vector field in U × C n that corresponding to the matrix A(x), then X = ∂ ∂x + v A (x). Let us compute the Lie bracket, Thus, Y is a symmetry if and only if dY Finally, by Lemma 4.1 we have that Y satisfies the stated differential equation if and only if it is a symmetry.
Corollary 6.1. Let x 0 be a non-singular point, and V a simply-connected neighbourhood of x 0 in U . Then, for each polynomial vector field Y (0) ∈ X[C n ] there is a unique vertically polynomial symmetry Y ∈ sym X (V) such that Y (x 0 ) = Y (0) . Moreover, sym X (V) and X[C n ] are isomorphic Lie algebras.
Proof. The map sym X (V) → X[C n ], Y → Y (x 0 ), is a Lie algebra morphism since the computation of the Lie bracket and the restriction to the fiber {x 0 } × C n are commuting processes. We have to see that it is an isomorphism. Let r be the degree of Y (0) and, r the decomposition of Y (0) in homogeneous components. Let Y k be the solution in V of the Cauchy problem: k . The existence and uniqueness of the solution guarantees that, is the only vertically polynomial symmetry such that Y (x 0 ) = Y (0)

Symmetries vs Galois
The intrinsic relation between the Galois group and the Lie algebra of symmetries of (5) is made explicit by the following result.
Theorem 6.2. Let Y (0) be a polynomial vector field in C n , and x 0 a non-singular point of (1). There is a vertically polynomial symmetry Y of X with coefficients in K such that Proof. It follows from our definition of Galois group. By Theorem 6.1, for each one of them there is a vertically polynomial Y symmetry such that Y (x 0 ) = Y (0) defined in a neigbourhood of x 0 . By theorem 6.1 this is a solution of a Lie-Vessiot system induced by X, here we consider all the homogeneous components simultaneously. Finally, by Lemma 5.1 this solution has coefficients in K if and only if A * Y (x 0 ) = Y (x 0 ) for all Galoisian matrices at x 0 .
Hence, the Galois group Gal(x 0 , X) of determines the Lie algebra sym <∞ (K) of vertically polynomial K-rational symmetries in the following sense. The Lie algebra sym <∞ (K) is isomorphic to X[C n ] Gal(x 0 ,X) , the Lie algebra of polynomial vector fields fixed by the action of Gal(x 0 , X) by linear transformations in C n . However, we do not have a reciprocal. In general, the Galois group is contained in the stabilizer of the Lie algebra of K-rational symmetries.
Let us check now how the vertically polynomial symmetries of different homogeneous degrees look like, and what kind information about the Galois group they carry.

Symmetries of degree zero
The canonical isomorphism X 0 [C n ] ≃ C n stated in Section 4.1 tell us that the linear representation (Φ 0 , X 0 [C n ]) is just an isomorphism. In particular, if This lemma can be understood as an infinitesimal version of the superposition principle. If y(x) and φ(x) are solutions of 1, then for all ε y(x) + εφ(x) is also a solution. For a fixed ϕ(x) and a general y(x), the derivative of this monoparametric family of solutions with respect to ǫ is a vertical vector field, namely, the symmetry Y .

Linear symmetries
Homogeneous vertically polynomial symmetries of degree one are called linear symmetries. The homogeneous vector field h = n i=1 y i ∂ ∂y i and its multiples gives us a trivial monoparametric family vertically linear symmetries for any system of differential equations. The canonical isomorphism X 1 [C n ] ≃ gl(n, C) stated in Section 4.1 tell us the that Lie-Vessiot sytem induced by the representation (Φ 1 , X 1 [C n ]) can be seen as a matrix equation. If we write: where B = (b ij ) stands for a n × n matrix of undetermined functions, the induced system is written as: This is the equation of isospectral deformations induced by A(x) and has been exhaustively studied. The set of rational solutions of (7) is called the eigenring, see [3] for an extensive study of its properties, notably to decompose linear differential systems. If B(x) is a solution of (7) it is well known that the Jordan canonical form of B(x) does not depend on the point x. Thus, given a linear symmetry Y with matrix B(x) we will classify it according to its Jordan canonical form: (a) If B(x) has at least two different eigenvalues we will say that is is a decomposer symmetry.
(b) If all the eigenvalues of B(x) are different we will say that Y is a complete decomposer symmetry.
(b) If the eigenspaces of B(x) are one-dimensional we say that Y is a solver symmetry. That means that its Jordan canonical form does not contain any block of the form: The following theorem is very close to some of the results of C. Jensen in [8] Section 9, on integration by quadratures, and results of M. A. Barkatou in [3] on decomposition -although we obtain it by different means and relate it with the linear symetries and the Galois group of the system.
where the spaces E i = ker(B − λ i Id) are the generalized eigenspaces of B. The group: is clearly conjugated to the group of block-diagonal matrices. Let us see that all Galoisian matrices at x 0 are in G. If σ is Galoisian then, σ * (Y (x 0 )) = Y (x 0 ), but that means σBσ −1 = B, so σ conjugates B with itself, and thus it sends generalized eigenspaces of B to themselves. (b)=⇒(a). Let us assume that Gal(x 0 , X) is conjugated to a sugbroup of the group of block-diagonal matrices. Then, we have a decomposition of C n in subspaces as in formula (8), such that for all σ ∈ Gal(x 0 , X), and index i = 1, . . . , r, σ(E i ) = E i . Let us consider the following linear vector fields h i in C n for i = 1, . . . , k defined by properties: where h stands for the homogeneous vector field, Let us consider µ 1 , . . . , µ k different complex numbers and define: By definition, Y (0) is stabilized by any Galoisian matrix, and them, by Lemma 5.1 there is K-rational symmetry Y whose value at x 0 is Y (0) . This symmetry Y is the decomposer symmetry of the statement.
Corollary 6.2. The following are equivalent: (a) The Galois group Gal(x 0 , X) is conjugated to subgroup of the group of diagonal matrices (C * ) n ⊂ GL(n, C).
(c) There is a complete-decomposer linear symmetry in sym 1 X (K). Proof. The statement is the particular case of Theorem 6.3 in which all the eigenvalues are simple.
Remark 6.1. The existence of a complete-decomposer linear symmetry implies the existence of a n-dimensional abelian Lie algebra of symmetries. Let us consider a completedecomposer linear symmetry Y and x 0 a non-singular point. Let B be the matrix of Y (x 0 ), and v 1 , . . . , v n be a basis of eigenvector of B. As before, we define vector fields: where h stands for the homogeneous vector field, It is easy to check that the matrices of the vector fields h i have common eigenvectors and then [ h i , h j ] = 0. For all Galoisian matrix σ at x 0 we have σ * ( h i ) = h i , and thus by Lemma 5.1 there are vertically linear K-rational symmetries H 1 , . . . , H n such that H i (x 0 ) = h i . They form a n-dimensional abelian Lie-algebra. In general, σ ker(B − λId) j = ker(σBσ −1 − λId) j thus, Galoisian matrices respect the chain of subspaces, in other words, they are triangular matrices in some suitable basis. For the general case, with different eigenvalues, we first consider the decomposition of the group by block-diagonal matrices given in Theorem 6.3, and then we apply the above argument.

Symmetries of higher degree
Given a polynomial vector field Y in C n a linear discrete symmetry of Y is a nondegenerated matrix σ such that σ * Y = Y . Here, σ stands for the transformation, y = (y 1 , . . . , y n ) → σy =   n j=1 σ 1j y j , . . . , n j=1 σ nj y j   .
If the expression in coordinates of Y is, then, the matrix σ is linear discrete symmetry of Y if and only if it satisfies the equations: P i (σy) = n j=1 σ ij P j (y).
If we are looking for the linear discrete symmetries of a homogeneous polynomial vector field of degree r, it yields a total of n × n+r−1 r . Thus, for generic polynomial vector fields of high degree the group of linear discrete symmetries reduces to the identity. in C 2 . Equations (9) for this particular case yield σ 2 21 y 2 1 + 2σ 21 σ 22 y 1 y 2 + σ 2 22 y 2 2 = σ 11 y 2 2 , 0 = σ 21 y 2 2 , equating each coefficient be obtain, σ 11 = σ 2 22 , σ 21 = 0, thus, the group of discrete linear symmetries is: The Lie-Vessiot induced system for vertically polynomial symmetries of arbitrary degree r is: We can consider all the homogeneous components simultaneously. Theorem 1 can be restated in the following terms: Corollary 6.3. Let Y (0) be a polynomial vector field in C n or arbitrary degree, and x 0 a non-singular point of equation (1). The necessary and sufficient condition for the existence of a vertically polynomial K-rational symmetry Y of X such that Y (x 0 ) = Y (0) is that the Galois group Gal(x 0 , X) is contained in the group of discrete linear symmetries of Y 0 .
Example 6.2. Let us consider the system: Where a(x), b(x), are arbitrary functions in K. A direct computation of the Lie bracket says that Y = y 2 1 ∂ ∂y 2 is a symmetry, and thus the Galois group of the equation (for any function field K) is contained in the group (10).