Isomorphism of Intransitive Linear Lie Equations

We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by E. Cartan.


Introduction
It is known [7] that the isomorphism of the fibers of transitive linear Lie equations at two points is sufficient to obtain the formal isomorphism of the Lie equations. This is proved by constructing a system of partial differential equations (SPDE) whose solutions would be these isomorphisms. This SPDE may be not integrable [2,11], although it is formally integrable. If the data are analytic, the SPDE is integrable.
In this paper we consider principally the extension of this theorem to intransitive linear Lie equations. Intransitive linear Lie equations generate a family of orbits on the manifold that we suppose locally to be a foliation. Given two intransitive Lie equations, consider the restriction of both Lie equations along two transversal to the orbits. If these restrictions are isomorphic in a certain sense, then we can construct a formally integrable SPDE whose solutions (if they exist) are isomorphisms of the two Lie equations. Therefore we prove, at least in the analytic case, that formal isomorphism of two linear Lie equations along transversal to the orbits can be extended locally to local isomorphism of the two linear Lie equations in a neighborhood of the two transversal.
Specifically, consider M and M ′ manifolds of the same dimension, V and V ′ integrable distributions of the same dimension on M and M ′ , respectively, N and N ′ submanifolds of M and M ′ such that each integral leaf of V and V ′ through points x ∈ M and x ′ ∈ M ′ intersect N and N ′ (at least locally) at unique points ρx and ρ ′ x ′ , respectively. Let be φ : N → N ′ a local diffeomorphism, a ∈ N , a ′ ∈ N ′ , φ(a) = a ′ , Q k φ the manifold of k-jets of local diffeomorphisms f : M → M ′ such that φρ(x) = ρ ′ f (x), and Q k φ the sheaf of germs of invertible local sections of Q k φ . Furthermore, let be R k ⊂ J k V , R ′k ⊂ J k V ′ intransitive linear Lie equations such that R 0 = J 0 V , R ′0 = J 0 V ′ . We say that R k at point a is formally isomorphic to R ′k at a point a ′ if we can construct a formally integrable SPDE S k ⊂ Q k φ such that any solution f of S k satisfies (j k+1 f ) * (R k ) = R ′k (see Definition 5.3). We prove in Proposition 5.3 that this condition is equivalent to the existence of F ∈ Q k+1 φ such that βF (a) = a ′ and F * (T M ⊕ R k ) = T M ′ ⊕ R ′k . Then we prove: Theorem 5.1. Let be Φ : N → Q k+1 φ such that βΦ = φ and Then given a diffeomorphism f : M → M ′ such that f * V = V ′ and f | N = φ there exists F ∈ Q k+1 φ satisfying F | N = Φ, βF = f and This theorem for transitive linear Lie equations is in [7]. The next step is to define an intransitive Lie algebra representing T N ⊕ R k | N . Transitive Lie algebras were defined in [10] as the algebraic object necessary to study transitive infinitesimal Lie pseudogroups. This program was pursued in the papers [6,7,8,9,14,16,17,20,24] disclosing the fitness of transitive Lie algebras to study transitive linear Lie equations.
At the same time, several tries were made to include intransitive linear Lie equations in this theory [12,13,18,19]. Basically, they associated a family of transitive Lie algebras on a transversal to the orbits with an intransitive infinitesimal Lie pseudogroup, each transitive Lie algebra corresponding to the transitive infinitesimal Lie pseudogroup obtained by restriction to each orbit of the infinitesimal intransitive Lie pseudogroup. Another approach was to study the Lie algebra of infinite jets at one point of the infinitesimal intransitive Lie pseudogroup; this algebra is bigraded, and this bigraduation may give the power series of the structure functions introduced in [1]. In the first approach, it may happen to be impossible to relate transitive Lie algebras along the transversal and give a continuous structure to this family, as the following example in [27] shows: let be Θ the infinitesimal intransitive Lie pseudogroup acting in the plane R 2 given by Θ = θ(x, y) ∂ ∂y such that ∂θ ∂y = a(x) ∂θ ∂x with a = 0, a(0) = 0 .
Two of these infinitesimal intransitive Lie pseudogroups given by functions a(x) andā(x) are isomorphic if and only if there exists a C ∞ -function b(x), with b(0) = 0 such thatā(x) = b(x)a(x). The restriction of Θ to the orbits {(x 0 , y) : y ∈ R}, with a(x 0 ) = 0 is the infinitesimal Lie pseudogroup of differentiable vector fields on R, and the restriction to the orbit where a(x 0 ) = 0 are the infinitesimal translations on R. This example shows that, even if the intransitive linear Lie equation associated to Θ has all properties of regularity, by choosing an appropriate function a(x), we obtain a highly discontinuous family of transitive Lie algebras along the transversal. If a(x) andā(x) have the same power expansion series around 0, the bigraded Lie algebras of infinite jets of vector fields of Θ andΘ at point (0, 0) are the same. So, the bigraded Lie algebras cannot distinguish between non isomorphic intransitive Lie pseudogroups. However,É. Cartan associated "structure functions" constant along the orbits with infinitesimal Lie pseudogroups. Therefore any definition of intransitive Lie algebras must contain a way of getting the germs of "structure functions" at the point considered.  [[ , ]] ∞ ) we can obtain the germ of R k | N at point a, and, by applying the Theorem of [25] and Theorem 5.1 of this paper, we get the germ of the linear Lie equation R k at the point a, up to a formal isomorphism (cf. Theorems 6.1 and 6.2 below).
We summarize the content of this paper. Section 2 presents basic facts on groupoids and algebroids of jets, the calculus on the diagonal introduced in [16,17], and the construction of the first linear and non-linear Spencer complexes. We tried to be as complete as possible, and the presentation emphasizes the geometric relationship between the first linear and non-linear Spencer operators, and the left and right actions of a groupoid on itself. We hope this section will facilitate the reading of this paper, since several formulas given here are sometimes not easily identified in [16,17,14,6,7], due to the simultaneous use of the first, second and the sophisticated non-linear Spencer complex, and the identifications needed to introduce them. Section 3 contains the construction of partial connections on J k V . These partial connections are fundamental for Section 5. Section 4 introduces the basic facts on linear Lie equations and the associated groupoids. Section 5 presents the definition of formal isomorphism of linear Lie equations, and the proof of Theorem 5.1. Section 6 introduces the definition of intransitive Lie algebras, and the notion of isomorphism of these algebras. The last section discuss the classification of intransitive linear Lie equations of order one in the plane, with symbol g 1 of dimension one. This classification contains the examples introduced above.

Preliminaries
In this section, we present some background material. The main references for this section are [16,17,14], and we will try to maintain the exposition as self-contained as possible, principally introducing geometrical proofs for actions of invertible sections of Q k+1 M on sections of T ⊕ J k T M .

Groupoids and algebroids of jets
Let be M a manifold and Q k M = Q k the manifold of k-jets of local diffeomorphisms of M . This manifold has a natural structure of Lie groupoid given by composition of jets and inversion where f : U → V , g : V → W are local diffeomorphisms of M , and x ∈ U . The groupoid Q k has a natural submanifold of identities I = j k id, where id is the identity function of M . Then we have a natural identification of M with I, given by x → I(x). Therefore we can think of M as a submanifold of Q k . There are two submersions α, β : Q k → M , the canonical projections source, α(j k x f ) = x, and target, β(j k x f ) = f (x). We also consider α and β with values in I, by the above identification of M with I.
There are natural projections π k l = π l : Q k → Q l , for k ≥ l ≥ 0, defined by π l (j k x f ) = j l x f . Observe that Q 0 = M × M and π k 0 = (α, β). The projections π k l commute with the operations of composition and inversion in Q k .
We denote by Q k (x) the α-fiber of Q k on x ∈ M , or Q k (x) = α −1 (x); by Q k (·, y) the β-fiber of Q k on y ∈ M , in another way, Q k (·, y) = β −1 (y) and Q k (x, y) = Q k (x) ∩ Q k (·, y). The set Q k (x, x) is a group, the so-called isotropy group of Q k at point x.
We denote by Q k the set of invertible sections of Q k . Naturally Q k has a structure of groupoid.
Similarly, we can introduce the groupoid of l-jets of invertible sections of Q k , and we denote this groupoid by Q l Q k . We have the inclusions If V k β ≡ ker β * ⊂ T Q k denotes the subvector bundle of β * vertical vectors, then the action (2.2) depends only on F (β(X)): In a similar way, F defines a right action which is a diffeomorphism The differentialF * ofF induces the right action where β * : T Q k → T M is the differential of β : Q k → M . We verify from (2.5) that the functionF restricted to the α-fiber Q k (y) depends only on the value of F in f −1 (y). If V k α = ker α * , then the right action (2.6) depends only on the value of F at each point, and the action (2.6) by restriction gives the action The vector fieldξ is determined by its restriction ξ to I.
Let be T = T M the tangent bundle of M , and T the sheaf of germs of local sections of T . We denote by J k T the vector bundle of k-jets of local sections of π : T → M . Then J k T is a vector bundle on M , and we also denote by π : This means we have a natural identification and if we denote by we have T Q k | I ∼ =J k T . Observe thatJ k T is a vector bundle on M . The restriction of β * : T Q k → T to T Q k | I , and the isomorphism T Q k | I ∼ =J k T defines the map which we denote again by β * . For more details on this identification and the map β * , see the Appendix of [14], in particular pages 260 and 274. If ξ is a section of J k T on U ⊂ M , let bē ξ(X) = ξ(β(X)) · X the right invariant vector field on Q k (·, U ). Thenξ hasF t , −ǫ < t < ǫ, as the 1-parameter group of diffeomorphisms induced by invertible sections F t of Q k such that Therefore, F 0 = I and Proposition 2.1. If f is a real function on U , and ξ, η sections of J k T on U , then Proof .
If J k T denotes the sheaf of germs of local sections of J k T , then J k T is a Lie algebra sheaf, with the Lie bracket [ , ] k .
where ξ k , η k ∈ J k T and f is a real function on M .
We denote by the same symbols as above the projections π k l = π l : J k T → J l T , l ≥ 0, defined by π l (j k x θ) = j l x θ. If ξ k is a point or a section of J k T , let be ξ l = π k l (ξ k ). The vector bundle J 0 T is isomorphic to T by the map β * : J 0 T → T , where β * (j 0 x θ) = θ(x), see (2.9). However, β * : J k T → T is not equal to π k 0 : J k T → J 0 T , but they are isomorphic maps. Again, we have the canonical inclusions, and we use the same notation as (2.1), for θ ∈ T . Analogously to the definition of holonomic sections of Q k , a section ξ k of J k T is holonomic if there exists ξ ∈ T such that ξ k = j k ξ. Therefore, if ξ k is holonomic, we have ξ k = j k (β * ξ k ).
If θ : Then ξ can be identified to the linear application x J k T is given by η = j 1 x µ, with µ(x) = θ(x), then (π) * (η − ξ)v = 0, and we remember that π : obtained in this way is exact, and we get an affine structure on J 1 J k T .
The linear operator D defined by is the linear Spencer operator. We remember that ξ k−1 = π k k−1 ξ k and The difference in (2.12) is done in J 1 J k−1 T and is in T * ⊗ J k−1 T , by (2.11).
The operator D is null on a section ξ k if and only if it is holonomic, i.e., Dξ k = 0 if and only if there exists θ ∈ T such that ξ k = j k θ.
For a proof, see [14]. The operator D extends to

The calculus on the diagonal
Next, following [16,17,14], we will relateJ k T to vector fields along the diagonal of M × M and actions of sections in Q k to diffeomorphisms of M × M which leave the diagonal invariant. We Then (ρ 1 ) * : H R −→T is an isomorphism, so we identify H R naturally with T by this isomorphism, and utilize both notations indistinctly.
Proposition 2.4. The Lie bracket in R satisfies: As f •ρ 1 is constant on the submanifolds {x}×M and ξ is tangent to them, we obtain ξ(f •ρ 1 ) = 0, and the proposition is proved.
A vector field in V R is given by a family of sections of T parameterized by an open set of M . Therefore there exists a surjective morphism The kernel of morphism Υ k is the subsheaf V R k+1 of V R constituted by vector fields that are null on ∆ at order k. Therefore R/V R k+1 is null outside ∆. It will be considered as a sheaf on ∆, and the sections in the quotient as sections on open sets of M . So the sheaf R/V R k+1 is isomorphic to the sheaf os germs of sections of the vector bundle T ⊕ J k T on M . So we have the isomorphism of sheaves on M , We usually denote byJ k T = T ⊕ J k T andJ k T = T ⊕ J k T . As the bracket on R induces a bilinear antisymmetric map, which we call the first bracket of order k, where Υ k (ξ) = ξ k and Υ k (η) = η k . It follows from Proposition 2.4 that [[ , ]] k satisfies: The following proposition relates [[ , ]] k to the bracket in T and the linear Spencer operator D in J k T . Proposition 2.5. Let be v, w, θ, µ ∈ T , ξ k , η k ∈ J k T and f ∈ O M . Then: where the bracket at right is the bracket in T ; where the bracket at right is the bracket in T .

Proof . (i) This follows from the identification of
Also by (2.14), we have Let beṼ R the subsheaf in Lie algebras of R such thatξ ∈Ṽ R if and only ifξ is tangent to the diagonal ∆.
We denote by J kT the subsheaf of T ⊕ J k T =J k T , whose elements arẽ since the vector fields inṼ R are tangents to ∆, it follows that the bracket inṼ R defines a bilinear antisymmetric map, called the second bracket, by and ν k : J kT → J k T is an isomorphism of vector bundles.
Proof . We will verify properties (i) and (ii) of Proposition 2.2. If θ, µ ∈ T , let be Θ, H ∈ V R as in the proof of Proposition 2.5. Then: Proof . It follows from Propositions 2.5 and 2.6.
As a consequence of Proposition 2.6, we obtain that is an isomorphism of Lie algebras sheaves, where the bracket in J kT is the second bracket [ , ] k as defined in (2.18), and the bracket in J k T is the bracket [ , ] k as defined in (2.10).
In a similar way, we obtain from (2.17) that we can define the third bracket as .
Proposition 2.7. The third bracket has the following properties: Proof . The proof follows the same lines as the proof of Proposition 2.6.
Let's now verify the relationship between the action of diffeomorphisms of M × M , which are ρ 1 -projectable and preserve ∆, on R, and actions (2.3) and (2.
As a special case, Let's denote by J the set of (local) diffeomorphisms of M × M that are ρ 1 -projectable and preserve ∆. We naturally have the application and from this it follows It is clear that It follows from definitions of J and R that the action is well defined. Then V andṼ are invariants by the action of J .
We have: since that So we proved By projecting, we obtain Observe that this formula depends only on σ k . (iii) By combining (i) and (ii), we obtain As ). By replacing this equality above we get It follows from Proposition 2.8 that action (2.20) projects on an action ( ) * : This action verifies It follows from Proposition 2.8 and (2.21) that (σ k+1 ) * (ξ k )(x) depends only on the value of σ k+1 (x) at the point x where ξ is defined, and (σ k+1 ) * (v)(x) depends on the value of σ k+1 on a curve tangent to v(x).
Item (ii) of Proposition 2.8 says that restriction to J kT of action (2.21) depends only on the section σ k , so the action In this case, we get and each ν k (σ k ) * ν −1 k acts as an automorphism of the Lie algebra sheaf J k T : If M and M ′ are two manifolds of the same dimension, we can define The groupoid Q k acts by the right on Q k (M, M ′ ), and Q ′ k = Q k (M ′ , M ′ ) acts by the left. Redoing the calculus of this subsection in this context, we obtain the analogous action of (2.21): This action also verifies (2.22).

The Lie algebra sheaf
In this subsection, we continue to follow the presentation of [16,17]. We denote by J ∞ T the projective limit of J k T , say, denotes the sheaf of germs of local sections of the vector bundle T Q k | I → I. From the fact that In the following, we use the notatioň We define the first bracket inJ ∞ T as: With the bracket defined by (2.24),J ∞ T is a Lie algebra sheaf. Furthermore, We extend now, as in [16,17,14], the bracket onJ ∞ T to a Nijenhuis bracket (see [3]) on We introduce the exterior differential d on We extend this operator to forms of any degree as a derivation of degree +1 The exterior differential d is linear, for ω ∈ ∧ r (J ∞ T ) * , and d 2 = 0. Remember that (ρ 1 ) * : T ⊕ J ∞ T → T is the projection given by the decomposition in direct sum ofJ ∞ T = T ⊕ J ∞ T . (We could use, instead of (ρ 1 ) * , the natural map α * : T ⊕ J k T → T , given by α * : T Q k | I → T , and the identification (2.8)). Then (ρ 1 ) * : T * → (J ∞ T ) * , and this map extends to (ρ 1 ) * : . . , r. It follows that d((ρ 1 ) * ω) = (ρ 1 ) * (dω). We identify ∧T * with its image in ∧(J ∞ T ) * by (ρ 1 ) * , and we write simply ω instead of (ρ 1 ) * ω.
Let be u = ω ⊗ξ ∈ ∧(J ∞ T ) * ⊗ (J ∞ T ), τ ∈ ∧(J ∞ T ) * , with deg ω = r and deg τ = s. We also define deg u = r. Then we define the derivation of degree (r − 1) and the Lie derivative which is a derivation of degree r. If v = τ ⊗η, we define A straightforward calculation shows that: In particular, we have the following formulas: Proof . It is a straightforward calculus applying the definitions.
If we define the groupoid then for σ = lim proj σ k ∈ Q ∞ , we obtain, from (2.21), is well defined. It follows from (2.22) that σ * :J ∞ T →J ∞ T is an automorphism of Lie algebra sheaf.

The first non-linear Spencer complex
In this subsection we will study the subsheaf ∧T * ⊗ J ∞ T and introduce linear and non-linear Spencer complexes. Principal references are [16,17,14].
Let be γ k the kernel of π k : J k T → J k−1 T . Denote by δ the restriction of D to γ k . It follows from Proposition 2.3(ii) that δ is O M -linear and δ : γ k → T * ⊗ γ k−1 . This map is injective, in fact, if ξ ∈ γ k , then by (2.11), δξ = −λ 1 (ξ) is injective. As Observe that we get the map , and, if we go on, we obtain the isomorphism where, given a basis e 1 , . . . , e m ∈ T with the dual basis e 1 , . . . , e m ∈ T * , we obtain the basis of S k T * ⊗ J 0 T , where k 1 + k 2 + · · · + k m = k, k 1 , . . . , k m ≥ 0 and l = 1, . . . , m. In this basis From the linear Spencer complex, we obtain the exact sequence of morphisms of vector bundles Let's now introduce the first non-linear Spencer operator D. The "finite" form D of the linear Spencer operator D is defined by Proposition 2.12. The operator D take values in T * ⊗ J ∞ T , so Proof . By applying (2.32) and (2.33), it follows that for ξ ∈ J ∞ T , and by replacing (2.39) in (2.38), we get (2.7)).
for v ∈ T .
Proof . It follows from (2.21) and Proposition 2.12 that Proposition 2.13. The operator D has the following properties: In particular, , with ξ ∈ J ∞ T , and σ t ∈ Q ∞ is the 1-parameter group associated to ξ, then Proof .
Proposition 2.12 says that D is projectable: It follows from [χ, χ] = 0 that If we define the non-linear operator then we can write (2.40) as The operator D 1 projects in order k to Here [[u, u]] k denotes the analogous of formulas (2.27) and (2.28) projected in the order k, so that the extension of first bracket makes sense. We will leave the details to the reader. We define the first non-linear Spencer complex by It is possible to define the first nonlinear Spencer complex D for invertible sections of Q ∞ (M, M ′ ) by: and the same formula of Proposition 2.12 holds: where σ = lim proj σ k ∈ Q ∞ (M, M ′ ). Other properties can easily be generalised.

Partial connections
In this section we will develop the concept of partial connections or partial covariant derivatives associated with the vector bundle H ⊕ J k V in the directions of the distribution V ⊂ T . We thank the referee for pointing out that this concept is already in [15, p. 24]. The construction of connections for J k T , the transitive case, is in [7]. Let be V an involutive subvector bundle of T , a ∈ M , and N a (local) submanifold of M such that T a N ⊕ V a = T a M . Then there exists a coordinate system (x, y) in a neighborhood of a such that a = (0, 0), the submanifolds given by points with coordinates x constant are integral submanifolds of V , and N is given by the submanifold y = 0. At least locally, we can suppose that the integral manifolds of V are the fibers of a submersion ρ : M → N . In the coordinates (x, y), we get ρ(x, y) = (x, 0). If we denote by H the subvector bundle of T given by vectors tangent to the submanifolds defined by y constant, then H is involutive and T = H ⊕ V . Also, T * = H * ⊕ V * . We denote by H and V the sheaves of germs of H and V , respectively.
We denote by Q k V the subgroupoid of Q k whose elements are the k-jets of local diffeomorphisms h of M which are ρ-projectable on the identity of N . In the coordinates (x, y), h(x, y) = (x, h 2 (x, y)). The sheaf of germs of invertible local α-sections of Q k V will be denoted by Q k V . The algebroid associated with Q k V is J k V , and we denote by J k V the sheaf of germs of local sections of J k V . Then the first non-linear Spencer operator D can be restricted to Q k+1 V , and the linear Spencer operator D can be restricted to J k+1 V, If d is the exterior differential, we get the decomposition d = d H + d V . The fundamental form χ decomposes in χ = χ H + χ V , where χ H (u) = χ(u H ) and χ V (u) = χ(u V ). The linear Spencer operator D also decomposes in D = D H + D V , and it follows from Proposition 2.11(iii) that and for ξ ∈ J k T .
Proposition 3.1. If ω ∈ ∧V * , and u ∈ ∧V * ⊗ J ∞ T , then: is a derivation of degree 1, it is enough to prove (i) for 0-forms f and 1-forms ω ∈ V * . From (2.26) we have It follows from Proposition 2.9(i) that (ii) By applying Proposition 2.9(iii), we obtain (iii) It follows from (2.28) that It is enough to prove [χ V , ξ] = D V ξ. It follows from Propositions 2.5(ii) and 2.9(ii) that From item (ii) of this proposition, (2.30) and (2.31) we obtain The first non-linear Spencer operator D also decomposes naturally in We obtain: and Proof . From hypothesis, we get f * (u H ) = (f * u) H and f * (u V ) = (f * u) V , and from equation (2.36), we get The proof of the second formula is analogous.
If we apply Proposition 3.2 to GF , we get and if we pose F = G −1 , we get where α ∈ ∧V * and |α| is the degree of α.
Let be ω ∈ V * ⊗ J k+1 V such that β * (i(w)ω) = w, for w ∈ V, and In the sequel, forξ ∈ H ⊕ J k V, we denote by [ω,ξ] k and [ω,ω] k+1 the analogous of formulas (2.27) and (2.28) projected in the order k and k+1, respectively, so that the same construction of third and second bracket make sense. We will use this convention in the present section when it makes sense, and leave the details to the reader.
Proof . If we apply formula (ii) of Proposition 2.9, we obtain If α ⊗ξ ∈ ∧V * ⊗ (H ⊕ J k V), we know from (2.28) that and from (2.26) that and as i(ξ)ω = 0, then Therefore, and from this it follows that Let be σ y : N → Q k+1 V a family of differentiable sections such that Proposition 3.4. The partial connection ∇ defined byω = χ V + ω, where ω is defined as in (3.4), is flat.
Proof . First of all, so ω satisfies the condition to define a partial connection. We will show thatω satisfies [ω,ω] k+1 = 0, i.e., the partial connection ∇ defined byω is flat. Let be, for v ∈ V, i(v)ω the right invariant vector field defined by To finish, we know that, for v, w ∈ V, i(v)ω, i(w)ω and i([v, w])ω are tangent to the submanifold σ(M ), and, as β so from Proposition 2.6, From Proposition 2.9(iii) we obtain where ω k+2 is a section in J k+2 V that projects on ω. The proposition is proved.
Let be Q k φ the sheaf of germs of invertible local sections of Q k φ . Then, by restriction of action (2.23), there exists an action of Q k+1 φ (similar to (2.21)) on T ⊕ J k V : and Denote by and if G ∈ Q k+1 φ −1 satisfy α(G) = β(F ), and for g = βG we have g * (H ′ ) = H, then by applying Propositions 3.2 and 3.6 to F G we get By posing G = F −1 we get and let be ω ′ and ∇ ′ as in (3.4) and Proposition 3.3, respectively. Following the proof of Proposition 3.5, take v ∈ V such that v is H-projectable, i.e., the 1-parameter group f t of v is given in coordinates by f t (x, y) = (x, h t (y)). Define F t ∈ Q k+1 V by F t (x, y) = σ ht(y) (x, 0)σ y (x, 0) −1 .
is the associated 1-parameter group of v ′ , and the 1-parameter group associated with ω ′ (v ′ ) satisfies i.e., which shows that Φ * (U + Ξ) is parallel with respect to ∇ ′ if and only if U + Ξ is parallel with respect to ∇. Taking account of (3.6), the equation (3.9) projected in order k is equivalent to, or, considering (3.7),

Linear Lie equations
Definition 4.1. Let be R k a subvector bundle of J k T . We define the prolongation R k+1 of R k by where the intersection is done in J 1 J k T .
We denote the prolongation (R k+1 ) +1 of R k+1 by R k+2 and so on, and by R k+l the sheaf of germs of local sections of R k+l , for l ≥ 0. Proposition 4.1. A section ξ ∈ J k+1 T is in R k+1 if and only if π k ξ ∈ R k and Dξ ∈ T * ⊗ R k .
It does not follow from this proposition that R k+l is a vector bundle, and that π k+l : R k+l → R k+l−1 is onto, for l ≥ 2. To obtain this, we need an additional condition. Definition 4.3. We say that the linear Lie equation R k is formally integrable if (i) R k+l is a subvector bundle of J k+l T , The symbol g k of R k is the kernel of π k−1 : R k → J k−1 T . Also, g k+l is the kernel of π k+l−1 : R k+l → R k+l−1 , for l ≥ 1. It follows from Proposition 4.1 and from (2.35) that we have the subcomplex for l ≥ 2.
Definition 4.4. We say that the symbol g k is 2-acyclic if the subcomplex (4.3) is exact for l ≥ 2.
A consequence of this proposition is: Given a linear Lie equation R k , let be the distribution B ⊂ T Q k defined by B X = R k β(X) .X, for X ∈ Q k . It follows from (2.10) and (4.1) that the distribution B is involutive. Let be P k (x) the integral leaf of B that contains the point I(x), and P k = ∪ x∈M P k (x). Then P k is a groupoid, and a differentiable submanifold at a neighborhood of I. As our problem is local, we will suppose that P k is a differentiable groupoid, the differentiable groupoid associated with the linear Lie equation R k . Then the linear Lie equation R k is the Lie algebroid associated with P k . As before, we denote by P k the groupoid of invertible sections of P k .
We define the prolongation P k+1 of P k by where λ 1 : Q k+1 → Q 1 Q k and Q 1 P k is the groupoid of 1-jets of invertible sections of P k . The following is Proposition 6.9(ii) of [17]: Then F ∈ P k+1 .
Proof . It follows from (2.37) where v ∈ T , so As i(v)DF ∈ R k , we get λ 1 F.(i(v)DF ) ∈ T P k . Also, we get from π k F ∈ P k that j 1 π k F.v ∈ T P k . Therefore, If the linear Lie equation R k is formally integrable, and P k is the differentiable groupoid associated with R k , it is true (cf. Proposition 6.1, [17]) that the prolongation P k+l of P k is the groupoid associated with the linear Lie equation R k+l . Therefore, π k+l : P k+l+1 → P k+l are submersions, for l ≥ 0.

Formal isomorphism of intransitive linear Lie equations
In the following sections, we consider intransitive linear Lie equations.
Definition 5.1. We say that a linear Lie equation R k ⊂ J k T is intransitive if there exists an integrable distribution V ⊂ T such that R k ⊂ J k V and π 0 (R k ) = J 0 V .
In reality, considering (4.1), we need only to verify that π 0 (R k ) is a subvector bundle of J 0 T . Our basic problem in this section is to determine the conditions for two intransitive linear Lie equations to be isomorphic. This means that there exists a diffeomorphism that sends one equation onto the other. In the sequel, we give a brief description of the system of partial differential equations that we should solve to obtain a class of diffeomorphisms f : M → M ′ such that (j k+1 f ) * (R k ) = R ′k . We utilize the same notation of Section 3. Consider R k ⊂ J k V and R ′k ⊂ J k V ′ intransitive linear Lie equations, and P k ⊂ Q k V and P ′k ⊂ Q k V ′ the associated groupoids.
Definition 5.2. We say that a submanifold S k ⊂ Q k φ is automorphic by P k if α : S k → M , β : S k → M ′ are submersions, and for every X ∈ S k (a, b), where a ∈ M and b ∈ M ′ , S k (·, b) = X • P k (·, a).
We denote by S k the set of invertible sections of S k .
Proposition 5.1. Let S k+1 be the prolongation of S k . Then an invertible section F ∈ Q k+1 φ is such that F (x) ∈ S k+1 (x) for every x ∈ α(F ) if and only if π k F ∈ S k and DF ∈ T * ⊗ R k .
Proof . The same proof of Proposition 4.4 applies.
We define the symbol The symbol g k S of S k is isomorphic to the symbol g k of R k , and we get an complex analogous to (4.3), and we define that g k S is 2-acyclic in the same way. From the formal integrability theorem (see [5]) we obtain: Then S k is formally integrable, and each prolongation S k+r is automorphic by P k+r , for r ≥ 1.
Definition 5.3. We say that the intransitive linear Lie equation R k ⊂ J k V is formally isomorphic to the intransitive linear Lie equation R ′k ⊂ J k V ′ at points a and a ′ , respectively, if there exists a diffeomorphism φ : N → N ′ , and a submanifold S k ⊂ Q k φ automorphic by P k and formally integrable, such that: If there exists a solution f : M → M ′ of S k , i.e., a diffeomorphism f such that j k f is a section of S k , and f (a) = a ′ , then R k at point a is said isomorphic to R ′k at point a ′ . This definition is essentially local. A most useful way to verify the formal isomorphism is given by proposition below, analogous of Proposition 5.1: are intransitive linear Lie equations, N and N ′ submanifolds of M and M ′ transversal to integral submanifolds of V and V ′ , respectively, and φ : N → N ′ a diffeomorphism, a ∈ N , a ′ ∈ N ′ , and φ(a) = a ′ . Suppose furthermore that the symbol g k of R k is 2-acyclic. If there exists F ∈ Q k+1 φ such that βF | N = φ, F * (R k ) = R ′k , and DF ∈ T * ⊗ R k , then R k at a is formally isomorphic to R ′k at a ′ .

Proof . Define
and α × β : S k+1 → U × U ′ is onto (at least locally). If S k = π k S k+1 , then it is a straightforward verification that S k is automorphic by P k and S ′k is automorphic by P ′k .
Given an invertible section G ∈ S k+1 , then in the neighborhood of each point of α(G), there are invertible sections G 1 ∈ P k+1 and G 2 ∈ P ′ k+1 such that G = G 2 F G 1 . In fact, given a point x ∈ α(G), there is an open set V x ⊂ α(G), with x ∈ V x , and an invertible section G 1 of P k+1 defined on V x , such that β(G 1 ) ⊂ α(F ). Let be G 2 = GG −1 1 F −1 , defined on f (β(G 1 )). Then, G 2 is an invertible section of P ′k+1 , and G| Vx = G 2 F G 1 . It follows from Proposition 2.13(i) and Proposition 5.1 that DG ∈ T * ⊗ R k on the open set V x . As the V x 's cover α(G), we get this property on all α(G). Therefore, S k is formally integrable, and conditions of Definition 5.3 are satisfied.
Corollary 5.1. Suppose that R k ⊂ J k V , R ′k ⊂ J k V ′ are intransitive linear Lie equations, N and N ′ submanifolds of M and M ′ , transversal to integral submanifolds of V and V ′ , respectively. Let be φ : N → N ′ a diffeomorphism, a ∈ N , a ′ ∈ N ′ , and φ(a) = a ′ . Suppose furthermore that the symbol g k of R k is 2-acyclic. If there exists F ∈ Q k+1 φ such that βF | N = φ, and F * (T ⊕ R k ) = T ′ ⊕ R ′k , then R k at a is formally isomorphic to R ′k at a ′ .
Let's now show the existence of a flat partial connection that leaves R k invariant.
Proposition 5.4. Let be R k ⊂ J k V an intransitive linear Lie equation. Then there exists a flat partial connection such that, restricted to H ⊕ R k , it satisfies Proof . Choose a family of differentiable sections σ y introduced in (3.3) satisfying σ y (N ) ⊂ P k+1 . As P k+1 is the groupoid associated with R k+1 , the form ω defined by (3.4) belongs to V * ⊗ R k+1 , and the partial connection ∇ : , as a consequence of (4.2). The proof now follows from Propositions 3.4 and 3.5. Now we prove the fundamental theorem for formal isomorphism of linear Lie equations: Theorem 5.1. Suppose that R k ⊂ J k V and R ′k ⊂ J k V ′ are intransitive linear Lie equations, N and N ′ submanifolds of M and M ′ transversal to integral submanifolds of V and V ′ , respectively, and φ : N → N ′ a diffeomorphism. Suppose furthermore that there exists Φ : N → Q k+1 φ such that βΦ = φ, and Then given a diffeomorphism f : Proof . Let be families σ y : N → P k+1 , σ ′y ′ : N ′ → P ′k+1 , as in the proof of Proposition 5.4, and defining flat partial connections Observe that by Proposition 5.4, ω ∈ V * ⊗ R k+1 , and ω ′ ∈ V ′ * ⊗ R ′ k+1 . Redefine H ′ = f * H, if necessary, to obtain (x ′ , y ′ ) = f (x, y) = (a(x), b(y)), and define F ∈ Q k+1 φ by F (x, y) = σ ′y ′ (x ′ , 0)Φ(x, 0)σ y (x, 0) −1 .

From this and Corollary 2.3 we obtain
It follows from (3.11) that So, from π k ω ∈ R k , π k ω ′ ∈ R ′ k and (5.1) we get Combining this with (5.1) and (5.2), we get DF ∈ T * ⊗ R k and F * (R k ) = R ′k , and by Proposition 5.3 the theorem follows.
Corollary 5.2. Suppose that R k ⊂ J k V and R ′k ⊂ J k V ′ are intransitive linear Lie equations, that N and N ′ are submanifolds of M and M ′ transversal to integral submanifolds of V and V ′ , respectively. Let be φ : N → N ′ a diffeomorphism such that φ(a) = a ′ , where a ∈ N and a ′ ∈ N ′ . Suppose furthermore that the symbol g k of R k is 2-acyclic. If there exists Φ : N → Q k+1 φ such that βΦ = φ, and Φ * (T N ⊕ R k | N ) = T N ′ ⊕ R ′k | N ′ , then R k at point a is formally isomorphic to R ′k at point a ′ .
Proof . The corollary follows from Theorem 5.1 and Corollary 5.1.

Intransitive Lie algebras
In this section, we associate an intransitive Lie algebra with a germ of an intransitive linear Lie equation. This definition must generalize the definition of transitive Lie algebra, and incorporate the fact that we can reconstruct an intransitive linear Lie equation from its restriction to a transversal to the orbits, unless of formal isomorphism, as the Theorem of [25] and Theorem 5.1 above shows.
We continue, in this section, to suppose that R k ⊂ J k V is an intransitive linear Lie equation and g k is 2-acyclic. We remember that it follows from these hypotheses, see Corollary 4.1, that the prolongations R k+l of R k , l ≥ 1, satisfy: We also assume that R l = π l R k is a subvector bundle of J l V for every 0 ≤ l ≤ k − 1, in particular, R 0 = J 0 V . We denote by O N,a the R-algebra of germs at point a ∈ N of local C ∞ real functions on N . The bilinear antisymmetric map The map (ρ 1 ) * : L → Der O N,a is the canonical projection given by the direct sum, and (i) R ′k = π k R ′k+1 is a vector sub-bundle of J k V ; (ii) [[T ⊕ R ′ k+1 , T ⊕ R ′ k+1 ]] k+1 ⊂ T ⊕ R ′ k ; (iii) the truncated O N,a -intransitive R-Lie algebra associated with R ′k+1 is L k+1 = π k+1 L k+2 . Furthermore, if h k = {ξ ∈ L k |π k−1 ξ = 0} is 2-acyclic, then R ′k is formally integrable.
The definitions of L and D(V ) depend on the choice of the transversal N . Let's now introduce a notion of isomorphism inspired in Theorem 5.1 such that the intransitive Lie algebras obtained at point a taking different transversal submanifolds are isomorphic. We maintain the notation of Section 5.
Suppose that R k ⊂ J k V , R ′k ⊂ J k V ′ are formally integrable intransitive linear Lie equations, N and N ′ submanifolds of M and M ′ transversal to integral submanifolds of V and V ′ , respectively, and φ : N → N ′ a diffeomorphism, a ∈ N , a ′ ∈ N ′ , and φ(a) = a ′ . We denote also by φ the isomorphism of R-algebras φ : O N,a → O N ′ ,a ′ , defined by φ(f ) = f φ −1 . Let be Φ j+1 : N → Q j+1 φ α-sections such that φ = βΦ j+1 and π j Φ j+1 = Φ j for j ≥ 0. Put Φ = lim proj Φ j . We get maps If Φ * is an isomorphism, then L is said isomorphic to L ′ . Proposition 6.1. Suppose that R k ⊂ J k V is a formally integrable intransitive linear Lie equation, a, b points of M , N , N 1 transversal to the orbits of R k through the points a, b, and L, L 1 the intransitive Lie algebras associated with R k at the points a, b (and transversal N , N 1 ), respectively. Let be ρ : M → N the fibration (at least locally) defined by the leaves of V . If ρ(a) = ρ(b), then the O N,a -intransitive Lie algebra L is isomorphic to the O N 1 ,b -intransitive Lie algebra L 1 .
Proof . If x, y ∈ M , ρ(x) = ρ(y), there exists X ∈ P m with α(X) = x, β(X) = y. Therefore, we can choose Φ j : N → P j such that βΦ j (N ) = N 1 . Then It follows from the formal integrability of P k that we can choose the family {Φ j : j ≥ 1} such that Φ j+1 projects on Φ j , for j ≥ 1. If Φ = lim proj Φ j , then Φ * L = L 1 .
With these definitions, we can state Corollary 5.2 as: Theorem 6.2. Suppose that R k ⊂ J k V is a linear Lie equation with symbol g k 2-acyclic, and R ′k ⊂ J k V ′ another linear Lie equation. Let be a ∈ M , a ′ ∈ M ′ , N and N ′ transversal to the orbits of R k and R ′k through the points a and a ′ , L k and L ′ k the truncated intransitive Lie algebras associated with R k and R ′k , at points a, a ′ and tranversal N and N ′ , respectively. If there exists Φ : N → Q k+1 φ such that βΦ = φ : N → N ′ , φ(a) = a ′ , and Φ * L k = L ′ k , then R k at point a is formally isomorphic to R ′k at point a ′ .

Application
As an application of this theory, we could utilize the definition of intransitive Lie algebras to obtain the intransitive linear Lie equations in the plane obtained byÉ. Cartan in [1]. We will be limited to classifying the first order intransitive linear Lie equations, with dim g 1 = 1. This will include the example we presented in the introduction, which was not presented by Cartan in his table, suppressed by a nullity hypothesis.
Let be V a 1-dimensional distribution on R 2 , which we can suppose is generated by the vector field ∂ ∂y . We will use the coordinate system p j,l , j, l ≥ 0, 0 ≤ j + l ≤ k, in J k V , defined by Then and if Y ∈ J k V is such that π 0 (Y ) = j 0 ∂ ∂y , then Y, f j,l ⊗ j 0 ∂ ∂y k = f j,l−1 ⊗ j 0 ∂ ∂y ∈ g k−1 V .
In the classification of [1], case 2 is represented only by β = 0.

Conclusion
The results of this paper show that the intransitive Lie algebra here introduced to represent a linear Lie equation at a point is sufficient to guarantee the existence and formal isomorphism of intransitive linear Lie equations. This brings a new way to pursue the study of intransitive Lie groups and the applications envisaged by Sophus Lie on the integrability of partial differential equations with a pseudogroup of invariants. It is clear that several problems can still exist, as the relationship between subalgebras of transitive algebras and intransitive algebras, and the notion of equivalence of intransitive algebras. A very interesting problem is the classification of simple intransitive Lie groups, since Cartan, in his list, excluded some classes of simple intransitive Lie group, as the example presented above.