Inverse Jacobi multiplier as a link between conservative systems and Poisson structures

Some aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of view, due to the Flow-Box Theorem we restrict ourselves to neighborhoods of singularities. In this sense, we characterize Poisson structures around the typical zero-Hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non-vanishing first jet by connecting with the classical Poincar\'e center problem. From the global point of view, we connect the property of being strictly conservative (the invariant measure must be positive) with the existence of a Poisson structure depending on the phase space dimension. Finally, weak conservativeness in dimension two is introduced by the extension of inverse Jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed. Examples including Lotka-Volterra systems, quadratic isochronous centers, and non-smooth oscillators are provided.


Introduction
The presence of finite-dimensional Poisson systems (see [29,32] and references therein for an overview of the classical theory) is ubiquitous in many branches of physics and applied mathematics. The specific format of Poisson systems has allowed the development of many tools for their analysis (for instance, see [15]- [17], [19]- [21] and references therein for a sample). In addition, Poisson dynamical systems are significant due to several reasons. One is that they constitute a generalization of classical Hamiltonian systems comprising nonconstant and degenerate structure matrices, as well as odd-dimensional vector fields (in contrast to classical Hamiltonian systems, which are always even dimensional). Additionally, the Poisson system format is not limited by the use of canonical transformations, since every diffeomorphic change of variables maps a Poisson system into another Poisson system.
Let us consider a smooth vector field having a finite-dimensional Poisson struc- of dimension n and rank r ≤ n constant in a domain (open and simply connected set) Ω ⊆ R n . Here J (x) and H(x) are the associated structure matrix and Hamiltonian function, respectively. Then under these hypothesis for each point x 0 ∈ Ω there is (at least locally in a neighborhood Ω 0 ⊂ Ω of x 0 ) a complete set of functionally independent Casimir invariants {D r+1 (x), . . . , D n (x)} in Ω 0 , as well as a transformation x → Φ(x) = y where Φ is a smooth diffeomorphism in Ω 0 bringing the system (1) into its Darboux canonical form. Thus, beyond the fact that Poisson systems are a formal generalization of classical Hamiltonian flows, Darboux Theorem provides the dynamical basis for such a generalization. Conservative dynamical systems are those that preserve a finite measure equivalent to a generalized volume. Classical Hamiltonian systems are important examples of conservative systems. Since classical Hamiltonian systems are also a particular case of Poisson systems, it is thus natural that many Poisson systems are also conservative, and conversely that many conservative systems are Poisson systems (but not necessarily Hamiltonian). In spite that the connection between both Poisson systems and conservative flows exists, none of them implies the other, and such link seems to remain relatively unexplored in the literature, at least to the authors' knowledge. The investigation of some aspects of such relationship is the leitmotiv of this work.
More precisely, we say that a C 1 vector field Y = n i=1 f i (x)∂ x i defined on Ω ⊂ R n is conservative if there is a non-negative integrable scalar function V nonidentically vanishing on any open subset of Ω such that the volume integral is preserved under the flow, that is, where Γ is any measurable subset of Ω and ϕ t (x) denotes the associated flow to Y. Various versions of the following result can be found in books such as [27] and [33].
Proposition 1. The C 1 function V : Ω → R which is non-identically vanishing on any open subset of Ω satisfies (2) on any measurable subset Γ ⊂ Ω if and only if V is a solution of the following linear partial differential equation where divY = n i=1 ∂ x i (f i (x)) is the divergence of the C 1 vector field Y. Any real C 1 non locally null function V in Ω satisfying (3) is called inverse Jacobi multiplier. In 1844 C.G.J. Jacobi introduced in the literature the nowadays called Jacobi (last) multiplier 1/V . Initially it was mainly used to find the last additional first integral needed to achieve complete integrability of Y. Later, S. Lie found some relationships between V and Lie point symmetries of Y. Recently, it has been proved that the existence of V implies severe consequences to the dynamics of Y on Ω. In particular, the invariant zero-set V −1 (0) contains, under some assumptions, orbits which are relevant in the phase portrait of Y such as periodic orbits, limit cycles, stable, unstable and center manifolds, etc. (see [3,8] for details).
We will use the following general lemma. Lemma 2. Any C 1 vector field Y is divergence free if and only if it has the constant inverse Jacobi multiplier V (x) = 1.
Conversely, assume now div(Y) ≡ 0. Then any inverse Jacobi multiplier V of Y satisfies Y(V ) = 0, and clearly V (x) = 1 is a solution of the equation.
From the point of view of the relationship with Poisson systems and their diffeomorphic transformation properties, it will be useful for us to know how inverse integrating factors change under orbital equivalence of vector fields, see [3,8] for further details.
Proposition 3. Let Φ be a diffeomorphism in Ω ⊂ R n with non-vanishing Jacobian determinant J Φ on Ω and let η : Ω → R be such that η ∈ C 1 (Ω) and η(x) = 0 everywhere in Ω. If V is an inverse Jacobi multiplier of the C 1 -vector field Y in Ω then η(V • Φ)/J Φ is an inverse Jacobi multiplier of the orbitally equivalent vector field η Φ * (Y).
The structure of the article is the following. Section 2 is devoted to the relationship between three-dimensional conservative and Poisson systems around a zero-Hopf singularity. In Section 3 the concept of strict conservativeness is introduced and its consequences for the existence of a Poisson structure are developed for general n-dimensional flows. To conclude, in Section 4 a theory of weak conservativeness for planar flows is outlined.

Characterizing Poisson structures around a zero-Hopf singularity
In a neighborhood of a regular point, due to the Flow-Box Theorem, any analytic vector field is both Poisson and conservative. Hence, from the local point of view, we shall restrict ourselves to neighborhoods of singular points. In agreement with the result in [1], in this section we shall focus on 3-d Poisson systems that we shall name generic: given a 3 × 3 structure matrix of constant rank 2 in the domain Ω ⊂ R 3 , such Poisson structure is called generic if there exists one Casimir invariant globally defined in Ω. (Note that in some cases, often related to non-holonomic dynamics, it is possible that a 3-d Poisson structure of constant rank 2 in Ω is not generic in such domain, see [4]- [7], [24] and references therein for further details).
The following preliminary result is required: Lemma 4. An analytic vector field Y in an open set Ω ⊆ R 3 is a generic Poisson system if and only if it is analytically completely integrable in Ω. In such case it can be written as Y(x) = η(x) (∇H 2 (x) × ∇H 1 (x)) where H 1 and H 2 are independent first integrals and η is an inverse Jacobi multiplier of Y. If η is a constant then div(Y) ≡ 0.
Proof. Clearly in Ω ⊆ R 3 any generic Poisson system is analytically completely integrable since it possesses two functionally independent analytic first integrals in Ω, namely the Hamiltonian and one Casimir.
Conversely, assume that Y has two analytic independent first integrals H 1 and H 2 in Ω. Then it is obvious that it can be written as Y(x) = η(x) (∇H 2 (x)×∇H 1 (x)) where x ∈ Ω, ∇H i is the gradient of H i , η is an analytic scalar function in Ω and the symbol × denotes the cross product in R 3 . It is straightforward to check that actually such a Y is a Poisson vector field with Hamiltonian H 1 and structure matrix Actually H 2 becomes the Casimir of J .
The fact that η(x) is an inverse Jacobi multiplier of Y can be easily checked by direct evaluation.
Remark 5. It is worth emphasizing that the singular points x 0 ∈ Ω ⊂ R 3 where the Poisson vector field vanishes have a special nature. More specifically, since the rank of the structure matrix is assumed to be constant and equal to 2, we have J (x 0 ) = 0. Therefore, we focus on the points where ∇H(x 0 ) = 0, namely on the critical points of the Hamiltonian. In the particular case of Ω ⊂ R 3 and after diffeomorphically reducing the system to the Darboux canonical form, it can be seen that the eigenvalues associated to the singularity only can be of the form either {0, ±λ} or {0, ±iω}, with both λ and ω real numbers. In the particular case ω = 0 the singularity is called a zero-Hopf singular point. Consequently, this is the generic singularity that can be found in a neighborhood of phase-space completely foliated by periodic orbits.
The previous lemma allows developing the next result. The converse is also true if Z h has a family of first integrals depending analytically on h.
Proof. Since the origin is a zero-Hopf singularity of Y, its linear part has associated eigenvalues {0, ±iω} with i 2 = −1 and ω ∈ R\{0}. Performing a linear change of variables and rescaling the time to set ω = 1 we write the linear part of Y into real Jordan canonical form, that is, where the F j are real analytic functions in Ω only possessing nonlinear terms. Since the linear part of Y has two independent first integrals x 3 and x 2 1 + x 2 2 it is clear that D can be chosen in the form D(x) = x 3 + · · ·, where the dots denote higher order terms. Then the analytic diffeomorphism Φ = (Id 2 , D) in Ω (where Id 2 is the identity in R 2 ) is tangent to the identity and Φ * Y = (−y 2 +F 1 (y))∂ y 1 + (y 1 +F 2 (y))∂ y 2 whereF i are analytic nonlinear terms. By construction it is clear that Z h = (−y 2 +F 1 (y 1 , y 2 , h))∂ y 1 + (y 1 +F 2 (y 1 , y 2 , h))∂ y 2 is an analytic family of vector fields defined in a neighborhood of the origin in R 2 and with parameter values of h close to zero. We emphasize that Z h has a branch of singularities (y * 1 (h), y * 2 (h)) emerging from (y * 1 (0), y * 2 (0)) = (0, 0) with associated eigenvalues (λ 1 (h), λ 2 (h)) and (λ 1 (0), λ 2 (0)) = (i, −i). Clearly the above singularities are monodromic for |h| sufficiently small.
We now make use of the assumption that Y is a generic Poisson vector field in Ω. Then Y has an analytic first integral H in Ω functionally independent of D. Clearly this additional first integral can be selected as H(x) = x 2 1 + x 2 2 + · · · and the orbits near the origin of Y are closed since they are the intersection of the level sets of H and D(x) = x 3 + · · ·. Thus this H exists if and only if (y * 1 (h), y * 2 (h)) is a branch of center singularities of Z h and we prove the first part of the theorem.
Conversely, we assume that Z h has the branch (y * 1 (h), y * 2 (h)) of nondegenerate center singularities. Therefore Z h possesses a family of first integralsĤ(x 1 , is an analytically first integral of Y. Therefore Y is analytically completely integrable in a sufficiently small neighborhood Ω and from Lemma 4 it is a Poisson system in Ω.

Remark 7.
It is interesting to note that from the Poincaré-Dulac normal form theory there is an analytic diffeomorphism Ψ near the origin such that, when Z h has a center at the origin then Ψ * Z h becomes the vector field: with f (0, h) = 0. Moreover, this is a classical Hamiltonian vector field with Hamiltonian function: In short, which is the planar reduction of the Darboux canonical form of the Poisson system Y of Theorem 6.
Remark 8. Observe that, from the Implicit Function Theorem, under the conditions of Theorem 6 there is an analytic function φ( Clearly in practice we do not have the explicit expression of φ but we can compute enough terms of the Taylor expansion of φ at (x 1 , x 2 ) = (0, 0). This expansion will permit us to calculate a sufficiently large string of Poincaré-Liapunov constants associated to the branch of monodromic nondegenerate singularities (x * 1 (h), x * 2 (h)) of the vector field Z h in an algorithmic way and try to solve the associated center-focus problem.

Example: 3D Lotka-Volterra system
Let us now consider the quadratic Lotka-Volterra family: These are models of common use in mathematical biology for the description of population interactions. In addition equations (5) are of the Poisson type [28]. Using the Darboux's integrability theory it is easy to check that the full family possesses the inverse Jacobi multiplier Now we will use the classical procedure to obtain the additional first integral I 2 (x) of (5) using V (x) and I 1 (x). More specifically we compute a planar system after restricting (5) to the level sets {I 1 (x) = h}. This can be done substituting h into the two first components of (5) yieldinġ This planar system has the inverse Jacobi multiplier from where we can obtain the first integral I(x 1 , x 2 ; h) of it having the form We have proved that (5) under the parameter restrictions (6) is analytically completely integrable in the domain Ω, hence it is a Poisson system. In fact we have that the Casimir invariant is D(x) = I 1 (x), the Hamiltonian function is which can be easily deduced from I 1 and I 2 and the structure matrix is

Example 1 of Theorem 6
Consider a vector field having a zero-Hopf singularity at the origin. Systeṁ corresponds to case (i) in Theorem 1.4 of [18]. Note that system (8) has the analytic first integral Therefore ∂ x 3 D(0, 0, 0) = 1 = 0 and the level sets {x ∈ Ω : D(x) = h} of D are given by the graph of a function x 3 = φ(x 1 ; h). Hence the restriction of the vector field (8) to the level sets {D = h} is Z h whose expression iṡ We note that since φ(0; h) = h, the eigenvalues at the origin of this planar family are ±i √ 1 + bh and therefore are pure imaginary for small values of |h|. Additionally, for such values of h, family (10) has a center at the origin because it is a Hamiltonian family. Since the Hamiltonian depends analytically on h near h = 0 then, using Theorem 6 we deduce that family (8) has a Poisson structure around the origin.
We will complete this example by explicitly showing the Poisson structure of (8). First we compute the function Then the HamiltonianĤ(x 1 , x 2 ; h) of (10) iŝ (8) given, up to a multiplicative constant, by Let X be the associated vector field to (8). It follows that there exists a scalar function η : From the explicit expressions of D(x) and H(x), direct calculations yield Finally we obtain that (8) is a Poisson system with Hamiltonian H and structure matrix J (x) which comes from (4) with H 2 = D, that is,

Example 2 of Theorem 6
The quintic familyẋ has a zero-Hopf point at the origin and the first integral D(x) = x 3 − x 2 1 . Then the reduced vector field Z h to the level sets {D = h} is given by the planar quadratic familẏ having a singularity at the origin with eigenvalues ±i. Therefore, the conditions under which the origin becomes a center for family Z h are well known, see the seminal papers [22,23] and [2]. In short, Z h has a center for all h if and only if either B 1 = 0 or C = 0, in which case there is an analytic first integralĤ(x 1 , x 2 ; h) at (x 1 , x 2 ) = (0, 0) for each h. We observe that the former center cases are not always Hamiltonian (this situation only appears when C = 2B 1 +B 2 = 0). Consequently, if B 1 C = 0 then family (12) does not have a generic Poisson structure in any domain Ω ⊂ R 3 containing the origin.
In the analysis of the first center strata we let B 1 = 0. If B 2 = 0 then Z h = Z 0 is independent of h so the first integralĤ(x 1 , x 2 ) is also independent. When B 2 = 0 then (see [31]) The second center strata will be analyzed with parameter restrictions B 1 = 0 and C = 0. If moreover B 2 = 0 then (see again [31]) whereas when B 2 = 0 then We note that in any center case the first integralĤ(x 1 , x 2 ; h) is analytic with respect to h at h = 0. Therefore, from Theorem 6 we conclude that family (12) has a Poisson structure around the origin if and only if B 1 C = 0. In fact, the explicit construction of the Hamiltonian H(x) and structure matrix J (x) can be done in a way analogous to that of family (8).

Example 3 of Theorem 6
The polynomial family of sixth degreė has a zero-Hopf point at the origin and the Then the reduced vector field Z h to the level sets {D = h} is given by the planar cubic familẏ having a singularity at the origin with eigenvalues ±i. We have defined the parameter vector µ = (h, A, B, C, E) ∈ I × R 4 where I ⊂ R is a small neighborhood of the origin. We claim that family Z h does not have a center at the origin for any parameter value h ∈ I. To prove such claim we will see that the first focal value associated to the origin of Z h is not identically zero for any h ∈ I. We briefly recall here the theory of focal values, see for example [30]. Using the complex coordinate z = x 1 + ix 2 ∈ C with i 2 = −1, any planar family Z h can be written in the forṁ z = iz + F (z,z; µ) wherez = x 1 − ix 2 and F (z,z; µ) = R 1 2 (z +z), i 2 (z − z); µ + iS 1 2 (z +z), i 2 (z − z); µ . Finally we complement this complex differential equation with its complex conjugate. Denoting by w =z we arrive at the complex polynomial systemż = iz + F (z, w; µ),ẇ = −iw +F (z, w; µ).
Now we define the focus quantities g j (µ) ∈ R[µ] as those polynomials such that X µ (H) = j≥1 g j (µ)(zw) j+1 where X µ = (iz + F (z, w; µ))∂ z + (−iw +F (z, w; µ))∂ w is the complex vector field in C 2 and H(z, w; µ) = zw + · · · ∈ C[[x, w]] is a formal power series. It is known that Z h * has a center at the origin for a specific parameter value µ = µ * if and only if g j (µ * ) = 0 for all j ∈ N. Performing computations we find that g 1 (µ) = 1 Since for every parameter combination we have g 1 (µ) ≡ 0 for arbitrary h ∈ I we prove that family Z h cannot have a center at the origin for any h ∈ I and therefore, from Theorem 6 we know that family (13) has no generic Poisson structure around the origin.

Conservativeness and Poisson structure
First of all, notice that there exist conservative planar vector fields that do not have a Poisson structure and do not describe physically conservative dynamics. For example, the linear systemẋ with µ = 0, has a focus at the origin and non-negative inverse Jacobi multiplier V (x 1 , x 2 ) = x 2 1 + x 2 2 . Therefore system (15) is conservative in Ω = R 2 , which clearly does not correspond to a conservative flow in direct physical terms. Moreover this system is not of Poisson type since it does not have an analytical first integral in any neighborhood of the origin. This situation is not exceptional. For instance, in [3] the following example appears. Systeṁ has the inverse Jacobi multiplier V (x) = (x 2 1 + x 2 2 ) 2 in Ω = R 3 . Since V is nonnegative it is clear that the system is conservative in R 3 . The eigenvalues of the linearization at the origin are { 1 2 (1 ±i), 1} and therefore, see Remark 5, the system has not a Poisson structure in a neighborhood of the origin. We note that additionally the system also possesses another inverse Jacobi multiplier V 2 (x) = x 3 which implies the existence of the rational first integral These examples suggest that the formal definition of conservative flow is not sufficiently restrictive from the most standard physical perspective. However, this difficulty can be overcome in very simple terms just by imposing that the Jacobi multiplier be strictly positive (or strictly negative, since −V is an inverse Jacobi multiplier if and only if V is). It is clear that system (15) has a positive inverse Jacobi multiplier if and only if µ = 0 and thus the origin is a center and the flow becomes Hamiltonian. This simple example reflects what is going to be the actual general situation in the two-dimensional case.

Definition 9.
A conservative vector field is strictly conservative if it has a strictly positive invariant measure, that is, satisfying (2) with V > 0 in Ω.
The trivial examples of strictly conservative systems are the divergence free systems, see Lemma 2. The next theorem connects the property of being strictly conservative with the existence of a Poisson structure depending on the phase space dimension.
Theorem 10. Let Y be a smooth vector field in Ω ⊆ R n . Then: (i) In the case n = 2, if Y is strictly conservative with smooth invariant measure in a simply-connected domain Ω then it is a Poisson vector field in Ω.
(ii) Let V be an inverse Jacobi multiplier of Y in Ω ⊆ R 2 . Then the zero-set Provided Ω i is simply connected, the restricted field Y| Ω i is a Poisson system orbitally equivalent to the Darboux canonical form which can be constructed globally in Ω i .
(iii) In the case n = 3, if Y is strictly conservative in Ω it is not necessarily a generic Poisson vector field in Ω.
Proof. In the planar case (i), if the smooth vector field Y = P (x 1 , x 2 )∂ x 1 + Q(x 1 , x 2 )∂ x 2 is strictly conservative in Ω ⊂ R 2 with a smooth invariant measure given by dx 1 dx 2 /V (x 1 , x 2 ) then we can construct a first integral H of Y in Ω as the line integral along any curve connecting an arbitrarily chosen point (x 0 1 , x 0 2 ) and the point (x 1 , x 2 ) in Ω. We remark that this line integral might not be well-defined if Ω is not simplyconnected, which is not our case. Also, clearly, since V > 0 and smooth in Ω we find that H is smooth in Ω, and consequently Y is a Poisson vector field in Ω (see [10] for the last sentence). This proves statement (i).
The proof of (ii) is constructive. First we recall that since V satisfies . In what follows we restrict the analysis to one single simply connected Ω i . Since Then it follows that div(Y/V ) ≡ 0 in the set Ω i . Since Ω i is simply connected, there exists a smooth function H : Ω i → R given by (16) such that Y/V is a Hamiltonian vector field in Ω i with Hamiltonian H. This induces the noncanonical Poisson structure of Y in Ω i in terms of Hamiltonian H and structure matrix This completes the proof of (ii).
To prove (iii) a counterexample can be used in dimension n = 3. We will show a vector field Y in a sufficiently small neighborhood Ω ⊂ R 3 of a zero-Hopf singularity at the origin such that Y is strictly conservative in Ω but it is not a generic Poisson vector field in Ω. Let us consider the quadratic family of vector fields in R 3 , the three-dimensional vector field formed by the first components of (18). Using Theorem 1.5 of [18], we know that there is a neighborhoodÛ of the origin in R 3 completely foliated by periodic orbits ofŶ, including continua of equilibria as trivial periodic orbits, if and only if one of the following parameter conditions hold: On the other hand, in [14] it is proved thatÛ exists if and only ifŶ is completely analytically integrable, that is, there are two independent analytic first integrals inÛ. Now it is easy to check that there are vector fieldsŶ satisfying (19) that do not satisfy any of the conditions (A-E). The counterexample follows by taking one of these vector fields andΩ ⊂Û, since generic Poisson vector fields inÛ are in particular completely analytically integrable inÛ . This proves part (iii). This completes the proof of statement (iii).
Remark 11. Let x * ∈ ∂Ω i ⊂ V −1 (0) be a point of the boundary of Ω i . Observe that the rank of the Poisson structure matrix (17) vanishes on x * . Accordingly, the Darboux canonical form is not defined on such boundary, since there is no constant-rank neighborhood of x * . At the same time, it is worth recalling that the line integral defining the Hamiltonian H in (16) is not defined on any neighborhood of x * due to the vanishing of V on such point.

Remark 12.
Under the same hypotheses of statement (ii) of Theorem 10, if some Ω i is not simply connected then the same construction is valid for every simply connected subdomain of it. This implies that Ω i can be fully decomposed as the union of simply connected subdomains on which the Darboux canonical form can be globally constructed.

Example: Poisson structure of the quadratic isochronous centers
An isolated singular point of Y is said to be a center if every orbit in a punctured neighborhood of it is a periodic orbit. Additionally, it is said to be an isochronous center if every periodic orbit in such a neighborhood has the same period. For the class of planar quadratic vector fields having an isochronous center at the origin, Loud proved in [26] that after a linear change of coordinates and a constant time rescaling the system can be brought into four canonical forms. Use of statements (i) and (ii) of Theorem 10 can be made in a similar way on the four isochronous cases in order to construct their Poisson structure and invariant measures. For the sake of illustration only one of them will be analyzed here. For this purpose the following isochronous system Y is chosen: In [11] it is found the inverse Jacobi multiplier V (x 1 , x 2 ) = (3−16x 2 )(9−24x 2 +32x 2 1 ). The set V −1 (0) is composed by one straight line and one parabola which do not intersect. Thus Ω = R 2 has the natural partition given by From the previous discussion it is found that (20) is a Poisson vector field in each Ω i with the same structure matrix and Hamiltonian obtained after evaluation of the line integral (16). As anticipated in Remark 11, the structure matrix J becomes singular on V −1 (0) = ∪ 3 i=1 ∂Ω i and, in addition, H is smooth on each Ω i but it is not defined on V −1 (0).

Remark 13.
Let Y be a smooth Poisson vector field in Ω ⊆ R n with constant rank r. Then for every x 0 ∈ Ω there is a neighborhood Ω 0 ⊂ Ω of x 0 and a smooth diffeomorphism Φ in Ω 0 such that Φ * Y is written in the Darboux canonical form, hence as a Poisson system with symplectic structure matrix S (n,r) of dimension n and rank r. Since Darboux theorem is not constructive, sometimes in practice [19]- [21] only a more general structure matrix η S (n,r) can be reached for Ψ * Y under a different diffeomorphism Ψ. In this case an additional time rescaling is required to complete the Darboux reduction. Now it is easy to check that function η : Ω 0 → R is just an inverse Jacobi multiplier of the Darboux canonical form (1/η)Ψ * Y. This implies that V = η/J Ψ is an inverse Jacobi multiplier of the original Poisson system Y according to Proposition 3.

A theory of weak conservativeness
Non-smooth differential systems are natural models in many branches of science such as mechanics, electromagnetic theory, automatic control, etc (see for instance [12,13,25]). Therefore we conclude by generalizing the previous planar theory to vector fields with less regularity. The first step is to extend the definition of inverse Jacobi multiplier for a planar vector field Y as weak solutions of the partial differential equation (3). In this section we will restrict ourselves to C 1 vector fields Y = P (x, y)∂ x + Q(x, y)∂ y defined on simply connected domains Ω ⊂ R 2 having smooth boundary ∂Ω.
The well known test functions will be used to introduce the forthcoming Definition 14 which is our definition of weak solution of the partial differential equation (3) defining the classical (hence C 1 ) inverse Jacobi multipliers. Recall that a function ϕ : Ω → R is called test function if ϕ ∈ C 1 (Ω) and there is a compact set K ⊂ Ω such that the support of ϕ is included in K. The linear space of all the test functions in Ω is denoted by D(Ω).

Definition 14.
A function W : Ω ⊂ R 2 → R is a weak inverse Jacobi multiplier in Ω of the vector field Y = P (x, y)∂ x + Q(x, y)∂ y with integrable divergence in Ω provided W is integrable in Ω and verifies The next result gives a relationship between inverse Jacobi multipliers in the plane and weak inverse Jacobi multipliers.
Theorem 15. Let V : Ω ⊂ R 2 → R and Y a vector field in Ω both C 1 (Ω). Then V is a weak inverse Jacobi multiplier of Y in Ω if and only if it is an inverse Jacobi multiplier.
Proof. Let V be an inverse Jacobi multiplier of Y in Ω. Consequently we have for every ϕ ∈ D(Ω). Now taking into account the identities the integrand of (21) can be rewritten in the form Due to the additivity of the integral, equation (21) becomes for every ϕ ∈ D(Ω). The first integral of the previous expression vanishes. To see this, Green's theorem on the plane can be applied as follows where in the last step we have used the fact that ϕ ∈ D(Ω) and therefore ϕ(x, y) = 0 for all (x, y) ∈ ∂Ω. To conclude, after rearrangement of the second integral in (22), equation (22) becomes Then V is a weak inverse Jacobi multiplier after Definition 14, thus concluding the proof in one sense. Now if V is C 1 (Ω) then the converse holds by reversing the previous steps. The proof is thus complete.

Piecewise C k weak inverse Jacobi multipliers
The following definition is now introduced.

Definition 16.
Let Ω ⊂ R 2 be a domain with smooth boundary ∂Ω. A function f : Ω → R is termed piecewise C k with k ≥ 1 in Ω if there exists a smooth curve γ ⊂ Ω such that Ω 1 ∪Ω 2 ∪γ = Ω, where γ = ∂Ω 1 ∩∂Ω 2 and f verifies f ∈ C k (Ω 1 ∪Ω 2 ), but f ∈ C 1 (γ). Additionally, a planar vector field Y is piecewise C k in Ω if both components are piecewise C k in Ω with respect to the same curve γ.
The forthcoming theorem will be useful in the context of weak inverse Jacobi multipliers.
Theorem 17. Let Y be a planar vector field admitting a weak inverse Jacobi multiplier W piecewise C 1 in Ω for the curve γ ⊂ Ω and W ∈ C(γ). Then γ is an invariant curve of Y.
Proof. Since W is a weak inverse Jacobi multiplier for Y in Ω we have for all ϕ ∈ D(Ω), where W i = W | Ω i . Since W i ∈ C 1 (Ω i ), we can make use of the identity Since Y(W i ) = W i divY in Ω i , the previous identity can be written in Ω i as This allows writing (23) as Applying Green's theorem to the two previous integrals we have where in the last step we use that by definition of test function in Ω it is ϕ(x, y) = 0 for all (x, y) ∈ ∂Ω i \γ for i = 1, 2. The factor (−1) i+1 takes into account the fact that the line integral has an opposite sense of integration for ∂Ω 1 and ∂Ω 2 . Therefore condition (24) yields γ (W 1 − W 2 )ϕ(P dy − Q dx) = 0, for all ϕ ∈ D(Ω). Since W ∈ C(γ) we conclude that P dy − Q dx = 0 on γ.
Consequently γ is an invariant curve of Y.

Remark 18.
Some simple examples of weak inverse Jacobi multiplier W of Y in Ω are listed below: (i) Bounded piecewise C 1 in Ω functions W with respect to a curve γ ⊂ Ω. If additionally W | Ω i > 0 then 1/W is an invariant measure for Y in Ω.

Example: general Poisson systems
Let us now consider an example of statement (ii) of Remark 18. As already recalled along the article, a general planar Poisson system corresponds to the C ∞ class and has the form dx dt = η(x 1 , x 2 )S (2,2) · ∇H(x 1 , x 2 ) where S (2,2) is the 2 × 2 symplectic matrix. As usual, assume the system is defined in a domain Ω. The structure matrix J (x 1 , x 2 ) = η(x 1 , x 2 )S (2,2) has constant rank 2 in Ω if and only if η does not vanish in Ω. In this case, as indicated in Remark 13 (see also references therein) it is possible to construct the Darboux canonical form globally in Ω by means of a time reparametrization and the smooth function η is an inverse Jacobi multiplier of the Darboux canonical form (which is now a classical Hamiltonian flow). The opposite case arises (as mentioned in item(ii) of Remark 18) provided there is a curve γ ≡ {(x 1 , x 2 ) ∈ Ω : η(x 1 , x 2 ) = 0} leading to a partition Ω = Ω 1 ∪ Ω 2 ∪ γ, with γ = ∂Ω 1 ∩ ∂Ω 2 , and η| Ω i having opposite signs. Now the global Darboux reduction is not possible since J is not regular in Ω. Note that η is still an inverse Jacobi multiplier (and now |η| is a weak inverse Jacobi multiplier) of the Darboux canonical form in Ω, and such reduction can be now carried out separately on each subdomain Ω i . From the point of view of time rescalings, in each subdomain Ω i they will have opposite signs, namely in one subdomain the direction of time will be inverted in the Darboux reduction, while in the other it will not. Additionally, in this case γ is an invariant curve.