Picard–Vessiot Extensions of Real Diﬀerential Fields

. For a linear diﬀerential equation deﬁned over a formally real diﬀerential ﬁeld K with real closed ﬁeld of constants k , Crespo, Hajto and van der Put proved that there exists a unique formally real Picard–Vessiot extension up to K -diﬀerential automorphism. However such an equation may have Picard–Vessiot extensions which are not formally real ﬁelds. The diﬀerential Galois group of a Picard–Vessiot extension for this equation has the structure of a linear algebraic group deﬁned over k and is a k -form of the diﬀerential Galois group H of the equation over the diﬀerential ﬁeld K (cid:0) √− 1 (cid:1) . These facts lead us to consider two issues: determining the number of K -diﬀerential isomorphism classes of Picard–Vessiot extensions and describing the variation of the diﬀerential Galois group in the set of k -forms of H . We address these two issues in the cases when H is a special linear, a special orthogonal, or a symplectic linear algebraic group and conclude that there is no general behaviour.


Introduction
To a homogeneous linear differential equation defined over a differential field K with field of constants k, Picard-Vessiot theory associates a differential field extension L of K, differentially generated over K by a fundamental system of solutions of the equation, and with constant field equal to k, called a Picard-Vessiot extension for the given equation. When k is algebraically closed, Kolchin [18] established that a Picard-Vessiot extension for the given equation exists and is unique up to K-differential isomorphism. The differential Galois group DGal(L/K) is defined as the group of K-differential automorphisms of L and has the structure of a linear algebraic group defined over k.
For a homogeneous linear differential equation defined over a formally real differential field K with real closed field of constants k, Crespo, Hajto and van der Put proved in [11], the existence and unicity up to K-differential isomorphism of a formally real Picard-Vessiot extension, endowed with an ordering extending the one in K. We note that such a linear differential equation may also have Picard-Vessiot extensions which are not formally real fields. Our result was later generalized in [15], by using model-theoretic methods, to the case when K is a differential field of characteristic 0 such that its field of constants k is existentially closed in K for strongly normal extensions of K associated to logarithmic differential equations over K on algebraic groups over k. Let L be a Picard-Vessiot extension of a formally real differential field K with real closed field of constants k. Then the differential Galois group G = DGal(L|K) has the structure of a linear algebraic group defined over k (see Section 2). Differential Galois theory over non-algebraically closed field of constants has been developed by several authors, see [1,2,19]. The inverse problem in this setting has also been considered (see [4,13]). In particular, Dyckerhoff proved in [13] that every linear algebraic group over R is a differential Galois group over the field R(x) of rational functions.
In Picard-Vessiot theory over formally real differential fields, one may find phenomena which do not arise in the context of differential fields with algebraically closed field of constants. Given a linear differential equation L(Y ) := Y (n) + a n−1 Y (n−1) + · · · + a 1 Y + a 0 Y = 0 with a n−1 , . . . , a 1 , a 0 belonging to a formally real differential field K with real closed field of constants k, one may ask the following questions which do not have a counterpart in the case when the field of constants k is algebraically closed.
-How many K-differential isomorphism classes of Picard-Vessiot extensions are there for L(Y ) = 0?
-Are the corresponding differential Galois groups k-isomorphic?
Concerning the first question, as mentioned above, it is known that there is at least one Picard-Vessiot extension of K for L(Y ) = 0, which is a formally real field. For the second one, let us note that if L and L are Picard-Vessiot extensions of a formally real differential field K with real closed field of constants k for the same equation, and G = DGal(L/K), G = DGal(L /K) are the corresponding differential Galois groups, then we have G × k k G × k k, i.e., G and G are both k-forms of the group H = G × k k.
In this paper, we consider a formally real Picard-Vessiot extension L of K for a linear differential equation L(Y ) = 0 defined over K and the differential Galois group G = DGal(L|K).
We give an answer to the above questions in the case in which H := G × k k is a special linear, special orthogonal, or symplectic linear algebraic group. Let us note that it makes sense to start with such a Picard-Vessiot extension L/K since in [11,Proposition 3.3] Crespo, Hajto and van der Put proved that when the differential field K is real closed, given a connected semisimple linear algebraic group G defined over k, there exists a linear differential equation defined over K and a formally real Picard-Vessiot extension L|K for it such that G = DGal(L|K). The inspection of the different cases shows that there is no general pattern. The differential Galois group may be the same for all Picard-Vessiot extensions of L(Y ) = 0 or range over the whole set of k-forms of H. For a linear differential equation defined over a formally real differential field K the determination of its real differential group G gives more information on the behaviour of the solutions than the determination of the complexification H of G. For example, for k = R and K a field of real functions, the determination of G will give information on the existence of oscillating functions among the solutions of L(Y ) = 0 (see [9] and the earlier topological approach in [14]). It is then interesting to study how the real differential group DGal(L|K) varies as L runs over the K-isomorphism classes of Picard-Vessiot extensions.
We refer the reader to [8] for the topics on differential Galois theory, when the field of constants is algebraically closed, to [6] or [21] for those on formally real fields and to [20,24,25] for those on linear algebraic groups.

Preliminaries
For the reader's convenience, we recall the definitions of formally real field, real closed field and Picard-Vessiot extension.
Definition 2.1. A formally real field is a field which may be given an ordering compatible with the field operations. Equivalently, a field K is formally real if −1 is not a sum of squares in K.
A formally real field is real closed if it has no nontrivial algebraic extensions which are formally real fields. Equivalently, a field k is real closed if −1 is not a square in k and k √ −1 is algebraically closed.
We note that in the literature on real algebraic geometry "real field" is frequently used for "formally real field". A field of positive characteristic is not formally real. The field R of real numbers is the standard example of a real closed field. The field R(x) of rational functions and the field of formal Laurent series R(x) with derivation d/dx are examples of real differential fields with real closed field of constants.
Definition 2.2. Given a linear differential equation L(Y ) := Y (n) + a n−1 Y (n−1) + · · · + a 1 Y + a 0 Y = 0 defined over a differential field K, with field of constants k, a Picard-Vessiot extension of K for L(Y ) = 0 is a differential field extension L|K such that a) L is differentially generated over K by a full set of solutions of L(Y ) = 0; b) the field of constants of L is k.
Let us assume that K is a formally real differential field with real closed field of constants k, L(Y ) = 0 a linear differential equation defined over K and L|K a Picard-Vessiot extension for L(Y ) = 0. In this case, we note that the set DHom K (L, L(i)) of K-differential morphisms from L to L(i) is in bijection with the set DAut K(i) L(i) of K(i)-differential automorphisms of L(i). We define the differential Galois group of L|K as the set DHom K (L, L(i)) with the group structure obtained by transferring the one of DAut K(i) L(i) via the above bijection. The differential Galois group has the structure of a k-defined linear algebraic group (see [10,Proposition 4.1]). We note that the proof of the existence of a formally real Picard-Vessiot extension for a linear differential equation defined over a formally real differential field with real closed field of constants given in [10, Theorems 3.2 and 3.3 and Corollary 3.4] is not right. The remaining proofs in [10] are correct.
In the sequel, K will denote a formally real differential field with real closed field of constants k, L(Y ) = 0 a linear differential equation defined over K, L|K a formally real Picard-Vessiot extension for L(Y ) = 0 and G the differential Galois group of L|K. We want to determine the number of Picard-Vessiot extensions of K for L(Y ) = 0, up to K-differential isomorphism, and the differential Galois group of each of them.
The set of K-differential isomorphism classes of Picard-Vessiot extensions for L(Y ) = 0 is in one-to-one correspondence with the cohomology set H 1 k, G k , where k denotes the algebraic closure of k and G denotes the differential Galois group DGal(L/K). Indeed, we have a bijection between the set of K-differential isomorphism classes of Picard-Vessiot extensions for L(Y ) = 0 and the set of isomorphism classes of fiber functors ω : M ⊗ → vect(k), where M ⊗ denotes the Tannakian category generated by the K-differential module M associated to L(Y ) = 0 [11,Proposition 1]. In turn, this set of isomorphism classes of fiber functors is in bijection with the set of isomorphism classes of G-torsors, by [12,Theorem 3.2]. Finally, the set of isomorphism classes of G-torsors is in bijection with H 1 k, G k (see, e.g., [27,Lemma A.5.1]). If L is a Picard-Vessiot extension for L(Y ) = 0, L(i) and L (i) are Picard-Vessiot extensions of K(i) for L(Y ) = 0. Since the field of constants of K(i) is k, we have an isomorphism of differential fields f : L(i) → L (i), by the unicity of the Picard-Vessiot extension in the case when the field of constants is algebraically closed. The group Gal k|k acts on the set of isomorphisms from L(i) to L (i) by s(f ) = s • f • s −1 , for s ∈ Gal k|k , where we denote also by s the automorphisms of L(i) and L (i) induced by s.
x is a 1-cocycle from Gal k|k to G k , then the corresponding Picard-Vessiot extension corresponding to x is the subfield of L(i) fixed by the automorphism x(c) • c, by Galois descent theory (see [23, Chapter III, Section 1.3]). The Galois group Gal k|k acts on G k , by an involution leaving G invariant, hence the cohomology set H 1 k, G k depends on the k-form Let now G denote a linear algebraic group defined over a real closed field k and let H = G × k k. The set of k-forms of H is in one-to-one correspondence with the cohomology set Let us note that the classification of the k-forms of H is equivalent to the classification of the real forms of the corresponding complex group.
We consider the map induced by the morphism from G k to Aut G k sending an element g in G k to conjugation by g. When G is the differential Galois group of a Picard-Vessiot extension L of K for a linear differential equation For c the nontrivial element of Gal k|k , we write a = c(a), for a an element in k. For v = (a 1 , . . . , a n ) ∈ k n , we shall write v = (a 1 , . . . , a n ) and for M = (a ij ) a matrix with entries in k, M = (a ij ). We shall consider a linear algebraic group G defined over the real closed field k, such that H = G × k k is either a special linear group SL(n), a special orthogonal group SO(n) or a symplectic group Sp(n). We will then consider the real forms of each of these groups. We refer to [23] or [16] for their determination, to [7] or [26] for a more explicit description of them.
We will determine in each case the number of K-differential isomorphism classes of Picard-Vessiot extensions of K for L(Y ) = 0 and the differential Galois group for each class.
Let i denote a square root of −1 in k. For p, n integers with 0 ≤ p ≤ n, we define the n × n matrices

Forms of SL(n)
The real forms of SL(n), n ≥ 2, are 1) SL(n, k); 2) SL(n/2, H) if n is even, where H denotes the quaternion algebra over k; 3) SU n, k, h , where h is a nondegenerate hermitian form on k n .
For G each of these real forms and K a formally real differential field, with real closed field of constants k, we consider a linear differential equation L(Y ) = 0 of order n defined over K and a Picard-Vessiot extension L|K for L(Y ) = 0 such that L is formally real and DGal(L/K) G.

G = SL(n/2, H)
Let us denote by 1, I, J, K the basis elements of H. We recall that GL(n/2, H) embeds into GL n, k via the morphism (h ij ) → (µ(h ij )), where We denote by A n the matrix (a ij ) 1≤i,j≤n with 0 in all other cases. By [17, Chapter VII, Section 29, Corollary 29.4], we have H 1 k, G k k * /Nrd H n/2 . Since the norm of a quaternion is always positive, we obtain H 1 k, G k = 2 (see also [23, Chapter III, Section 1.4]). We have then two Picard-Vessiot extensions for L(Y ) = 0, up to Kdifferential isomorphism. A nontrivial 1-cocycle x of Gal k|k in SL n, k is given by x(c) = ζ Id, for ζ a primitive n-th root of unity. A K-differential automorphism of L(i) corresponding to x is given by the matrix A := ζ −1/2 Id on the vector space of solutions, since A satisfies [22, Chapter X, Section 2, Proposition 4]). Conjugation by ζ −1/2 Id leaves the group SL(n/2, H) stable. We obtain that the Picard-Vessiot extensions for L(Y ) = 0 in both K-differential isomorphy classes have SL(n/2, H) as differential Galois group.

G = SU n, k, h
It is known that if h is a nondegenerate hermitian form on k n , then h is equivalent to a hermitian form with matrix I p , for some integer p with 0 ≤ p ≤ n, called the index of h and that two nondegenerate hermitian forms on k n are equivalent if and only if their indices coincide (see [17, Chapter VII, Section 29, Example 29.19] and [5, Section 3.3]). We fix G = M ∈ SL n, k : M t I p M = I p and consider the action of Gal k|k on SL n, k given by c(M ) = I p M t −1 I p . We shall prove , when n is odd or p is even, when n is even and p is odd.
To this end we shall determine a maximal set of pairwise nonequivalent 1-cocycles from Gal k|k in SL n, k . For such a cocycle x, we may assume that the image of Id ∈ Gal k|k is the identity matrix and then x is determined by the image B ∈ SL n, k of the unique non trivial element c in Gal k|k . By the 1-cocycle condition, B must satisfy Bc(B) = Id. We denote by x q the cocycle given by c → B q , where B q = I q I p , with q an integer of the same parity as p and 0 ≤ q ≤ n. Let us see that every 1-cocycle x from Gal k|k in SL n, k is equivalent to some x q . As said above, such a cocycle x is determined by x(c) = B satisfying Bc(B) = Id, i.e., BI p B t −1 I p = Id, equivalently BI p = I p B t = BI p t , so BI p is an hermitian matrix, hence there exists an invertible matrix M such that M −1 BI p M t −1 = I r , for some integer r, 0 ≤ r ≤ n. Equivalently Let us note that, taking determinants in (3.3), we obtain det M det M = 1, hence there exists ζ ∈ k such that det(ζM ) = 1 and ζM satisfies (3.3). We have then that there exists a matrix M ∈ SL n, k satisfying (3.3), which means that x is equivalent to the 1-cocycle x r determined by c → I r I p . Since B ∈ SL n, k , we have det(I r I p ) = 1, so r is an integer of the same parity as p.
Let us see now that the 1-cocycles x q are pairwise nonequivalent. We have x q ∼ x q ⇔ ∃ M ∈ SL n, k such that B q = M −1 B q c(M ). This equality is equivalent to M I q M t = I q which implies q = q , so the 1-cocycles x q are pairwise nonequivalent. We have then H 1 k, G k = {[x q ] : 0 ≤ q ≤ n, q ≡ p (mod 2)}. Then H 1 k, G k = |{q ∈ Z : 0 ≤ q ≤ n, q ≡ p (mod 2)}| and we obtain the values in (3.2).
We have Z SL n, k = µ n k . We want to determine the image of [x q ] under the map Φ : H 1 k, G k → H 1 k, Aut G k . The 1-cocycle x q corresponds to a Picard-Vessiot extension L q of K for L(Y ) = 0 such that there is a differential isomorphism f q from L(i) to L q (i) satisfying x q = f −1 q c(f q ). The differential isomorphism f q is determined by the matrix D q giving the images of a vector space of solutions. The isomorphism f q satisfies x q = f −1 q c(f q ) if and only if the matrix D q satisfies B q = D −1 q D q . We may take D q := J q J p . Since conjugation by D q leaves the group G invariant, we obtain that all Picard-Vessiot extensions of K for L(Y ) = 0 have the same differential Galois group G.
Gathering the results in this section we may state the following theorem.
Theorem 3.1. Let K be a formally real differential field with real closed field of constants k, L(Y ) = 0 a linear differential equation defined over K, L|K a formally real Picard-Vessiot extension for L(Y ) = 0 and G the differential Galois group of L|K. We assume that G is a real form of SL(n).
(1) If G = SL(n, k), L|K is the unique Picard-Vessiot extension for the equation L(Y ) = 0, up to K-differential isomorphism.
(2) If G = SL(n/2, H), there are two Picard-Vessiot extensions for the equation L(Y ) = 0, up to K-differential isomorphism, and both of them have differential Galois group G.
(3) If G = SU n, k, h , there are [n/2] + 1 (resp. [n/2]) Picard-Vessiot extensions for the equation L(Y ) = 0, if n is odd or p is even (resp. if n is even and p is odd), up to K-differential isomorphism, and all of them have differential Galois group G.

Forms of SO(n)
The real forms of SO(n), with n odd, are the groups SO(n, k, Q), where Q is a nondegenerate quadratic form on k n . When G is one of these forms, we proved in [11, Section 3, Examples 1 and 3] that the map Φ : H 1 k, G k → H 1 k, Aut G k is a bijection. We consider now the case when n is even. The real forms of SO(n), with n even, are 1) SO(n, k, Q), where Q is a nondegenerate quadratic form on k n ; 2) SU(n/2, H, h), where h is a nondegenerate anti-hermitian form on H n/2 (with respect to the involution σ of H defined by a + bI + cJ + dK → a − bI − cJ − dK).
For G each of these real forms and K a formally real differential field, with real closed field of constants k, we consider a linear differential equation L(Y ) = 0 of order n defined over K and a Picard-Vessiot extension L|K for L(Y ) = 0 such that L is formally real and DGal(L/K) G.

G = SO(n, k, Q)
The quadratic form Q is equivalent to a quadratic form with matrix I p , for some integer p with 0 ≤ p ≤ n, which determines the equivalence class of Q.
The cohomology set H 1 k, G k is in one-to-one correspondence with the set of equivalence classes of quadratic forms on k n of rank n and index of the same parity as p (see [17, Chapter VII, Section 29, formula (29.29)]). We have then when p is odd.
The cocycles x q defined by c → B q , where B q = I q I p , with q an integer of the same parity as p and 0 ≤ q ≤ n, form a complete system of representatives of the cohomology set H 1 k, G k . The 1-cocycle x q corresponds to a Picard-Vessiot extension L q of K for L(Y ) = 0 such that there is a differential isomorphism f q from L(i) to L q (i) satisfying x q = f −1 q c(f q ). The differential isomorphism f q is determined by the matrix D q giving the images of a vector space of solutions. The isomorphism f q satisfies x q = f −1 q c(f q ) if and only if the matrix D q satisfies B q = D −1 q D q . We may take D q := J q J p . If the matrix M satisfies M t I p M = I p , the conjugate matrix N := D q M D −1 q satisfies N t I q N = I q , hence the Picard-Vessiot extension corresponding to the 1cocycle x q has differential Galois group SO(n, k, Q q ), where Q q denotes the quadratic form with index q. Let us note that SO(n, k, Q q ) = SO(n, k, Q n−q ), hence the Picard-Vessiot extension corresponding to x q and x n−q , 0 ≤ q ≤ (n/2) − 1, have the same differential Galois group. We consider the exact sequence

G = SU(n/2, H, h), h anti-hermitian
Since the reduced norm of a quaternion is always positive, the reduced norm U(n/2, H, h)(k) → µ 2 (k) is the trivial map. We obtain then for the cohomology sets the exact sequence Therefore H 1 k, SU(n/2, H, h) k = 2. We obtain then that there are two Picard-Vessiot extensions for L(Y ) = 0, up to K-differential isomorphism. We denote by L the non formally real one. We may check that µ(G) is the intersection of (a conjugate form of) SO n, k with µ(GL(n/2, H)). A nontrivial 1-cocycle of Gal k|k in G k is given by c → A n , for A n the matrix defined by (3.1). The matrix B = (b ij ) defined by satisfies B −1 c(B) = A n , hence the cohomology class [x] corresponds to the isomorphism class of the differential isomorphism f from L(i) to L (i) with matrix B on the vector space of solutions. Since conjugation by B leaves G invariant, we obtain that both Picard-Vessiot extensions have the same differential Galois group.
Gathering the results in this section we may state the following theorem. For completeness, we include the case n odd.
Theorem 4.1. Let K be a formally real differential field with real closed field of constants k, L(Y ) = 0 a linear differential equation defined over K, L|K a formally real Picard-Vessiot extension for L(Y ) = 0 and G the differential Galois group of L|K. We assume that G is a real form of SO(n).
If n is odd, there are (n + 1)/2 Picard-Vessiot extensions for the equation L(Y ) = 0, up to K-differential isomorphism, and their differential Galois groups range over the whole set of real forms of SO(n).
Assume n even.
(1) If G = SO(n, k, Q p ), where Q p is a nondegenerate quadratic form on k n , of index p, there are (n/2) + 1 (resp. n/2) Picard-Vessiot extensions for the equation L(Y ) = 0, up to Kdifferential isomorphism, when p is even (resp. when p is odd) and their differential Galois groups range over the whole set of groups G = SO(n, k, Q q ), with Q q a nondegenerate quadratic form on k n , of index q, 0 ≤ q ≤ n/2 and q of the same parity as p.
(2) If G = SU(n/2, H, h), where h is a nondegenerate anti-hermitian form on H n , there are two Picard-Vessiot extensions for the equation L(Y ) = 0, up to K-differential isomorphism, and they have both differential Galois group G.

Forms of Sp(2n)
The real forms of Sp(2n) are 1) Sp(2n, k); 2) SU(n, H, h), where h is a nondegenerate hermitian form on H n (with respect to the involution σ of H defined by a + bI + cJ + dK → a − bI − cJ − dK).
For G each of these real forms and K a formally real differential field, with real closed field of constants k, we consider a linear differential equation L(Y ) = 0 of order n defined over K and a Picard-Vessiot extension L|K for L(Y ) = 0 such that L is formally real and DGal(L/K) G.

G = SU(n, H, h), h hermitian
If h is a nondegenerate hermitian form on H n , then h is equivalent to a hermitian form with matrix I p , with p ≥ n − p [7, Section 7.5.3]. The number of equivalence classes of nondegenerate hermitian forms over H n is then n 2 + 1. We fix G = M ∈ GL(n, H) : σ(M ) t I p M = I p .
The group G is the group of automorphisms of the hermitian vector space H n , h . Hence the set of equivalence classes of nondegenerate hermitian forms over H n is in one-to-one correspondence with the cohomology set H 1 k, G k (see [22, Chapter X, Section 2, Proposition 4] or [16, Section 2.6, Lemma 3]). Since the set of K-differential isomorphism classes of Picard-Vessiot extensions for L(Y ) = 0 is also in one-to-one correspondence with the cohomology set H 1 k, G k , we have that the number of K-differential isomorphism classes of Picard-Vessiot extensions for L(Y ) = 0 is equal to the number of equivalence classes of nondegenerate hermitian forms over H n . Let us note that for M, N ∈ M n (H), we have σ(M N ) t = σ(N ) t σ(M ) t . We determine now the image of G under the isomorphism µ ⊗ k k : GL n, H ⊗ k k → GL 2n, k .
2n , for A 2n the matrix (a ij ) 1≤i,j≤2n defined by 0 in all other cases.
is a conjugate form of Sp 2n, k and µ(G) = N ∈ G : N = A 2n N A −1 2n . A complete set of nonequivalent 1-cocycles of Gal k|k in µ(G) is given by with q an integer, 0 ≤ q ≤ n, q ≥ n − q. The 1-cocycle x q corresponds to a Picard-Vessiot extension L q of K for L(Y ) = 0 such that there is a differential isomorphism f q from L(i) to L q (i) satisfying x q = f −1 q c(f q ). The differential isomorphism f q is determined by the matrix D 2q giving the images of a vector space of solutions. The isomorphism f q satisfies x q = f −1 q c(f q ) if and only if the matrix D 2q satisfies B q = D −1 2q D 2q . We may take D 2q := J 2q J 2p . If N is a matrix belonging to µ(G), it satisfies N t A −1 2n I 2p N = A −1 2n I 2p . Then the conjugate matrix P of N by D 2q , P := D 2q N D −1 2q , satisfies P t A −1 2n I 2q P = A −1 2n I 2q , hence the Picard-Vessiot extension corresponding to the 1-cocycle x q has differential Galois group SU(n, H, h q ), where h q denotes the hermitian form with index q.
Gathering the results in this section we may state the following theorem.
Theorem 5.1. Let K be a formally real differential field with real closed field of constants k, L(Y ) = 0 a linear differential equation defined over K, L|K a formally real Picard-Vessiot extension for L(Y ) = 0 and G the differential Galois group of L|K. We assume that G is a real form of Sp(2n).
(2) If G = SU(n, H, h p ), where h p is a nondegenerate hermitian form on H n , of index p, 0 ≤ p ≤ n, p ≥ n − p, there are [n/2] + 1 Picard-Vessiot extensions for the equation L(Y ) = 0, up to K-differential isomorphism, and their differential Galois groups range over the whole set of groups G = SU(n, H, h q ), with h q a nondegenerate hermitian form on H n , of index q, 0 ≤ q ≤ n, q ≥ n − q.

Conclusions
In the preceding we have seen cases in which a linear differential equation L(Y ) = 0 defined over a formally real differential field K has Picard-Vessiot extensions which are not formally real.
The occurrence of these extensions depends on the real form of the differential Galois group of L(Y ) = 0. When the number of K-differential isomorphisms of Picard-Vessiot extensions of L(Y ) = 0 is bigger than 1, we find several situations concerning the differential Galois group, either it is the same for all Picard-Vessiot extensions or it ranges over a subset or the whole set of real forms of the group G of the formally real Picard-Vessiot extension. It would be interesting to know if, in the case when K is a field of real functions, the solutions of such an equation in a non formally real differential field and the variation of the differential Galois group have some physical interpretation. Some inspiring examples in Hamiltonian mechanics are presented in [3].