NON-FORMALLY INTEGRABLE CENTERS ADMITTING AN ALGEBRAIC INVERSE INTEGRATING FACTOR

. We study the existence of a class of inverse integrating factor for a family of non-formally integrable systems whose lowest-degree quasi- homogeneous term is a Hamiltonian vector ﬁeld. Once the existence of an inverse integrating factor is established, we study the systems having a center. Among others, we characterize the centers of the perturbations of the system − y 3 ∂ x + x 3 ∂ y having an algebraic inverse integrating factor.

1. Introduction. One of the classical problems in the qualitative theory of the analytic planar systems is to characterize when a monodromic point (singular point which is surrounded by orbits of the system) is a center, usually called elliptic point for integrable dynamical systems in dimension higher than two. For planar vector fields, this problem (so-called center problem) was solved theoretically for a nondegenerate singular point (systems whose linear part evaluated at the singular point have two imaginary nonzero eigenvalues) and for the nilpotent case. The problem is still unsolved for the remaining case, i.e. the systems with linear part identically zero at the singular point, called degenerate singular point.
One of the main tools used to characterize non-degenerate and nilpotent centers is the computation of a normal form [27,28]. Normal forms for non-degenerate points are also of interest in symplectic geometry [10,15,16]. For the degenerate case, it is not surprising that a possible solution might be given by means of the theory of normal forms.
Another question related to the center problem is, once the monodromy is established, to determine the existence of an analytic first integral. Thus, for instance, for a non-degenerate singular point, the analytic integrability and the center problem are equivalent. Namely, the vector field (−y+· · · )∂ x +(x+· · · )∂ y has a center at the origin if, and only if, it has an analytic first integral of the form x 2 +y 2 +· · · . Otherwise, for nilpotent or degenerate singular points, in order to determine whether the singular point is a center, the existence of a first integral is a sufficient condition, but not a necessary one.
In this context, the existence of an integrating factor or an inverse integrating factor enables us to provide information about both center and integrability problems.
We consider an autonomous systeṁ where F is an analytic planar vector field defined in a neighborhood of the origin Ω ⊂ C 2 having a singular point at the origin, i.e. F(0) = 0 and P, Q ∈ C[[x, y]] (algebra of the power series in x and y with coefficients in C).
A non-null C 1 class function V is an inverse integrating factor of the system (1) (or also of F) on Ω if it satisfies the linear partial differential equation L F V = div(F)V, with L F V := P ∂V /∂x + Q∂V /∂y, the Lie derivative of V with respect to F, and div(F) := ∂P/∂x + ∂Q/∂y, the divergence of F. This name for V comes from the fact that V −1 defines on Ω \ {V = 0} an integrating factor of the system (1), i.e. F/V is divergence-free. So, if the system (1) possesses a formal inverse integrating factor V then it is formally integrable on Ω \ {V = 0}.
The existence of an inverse integrating factor has been used to study the center problem for non-degenerate singular point [26], the center problem and analytic integrability for nilpotent vector fields [7], the Darboux integrability [12,13,25], the Hopf bifurcation [19] and the existence of Lie symmetries [24].
The inverse integrating factor also plays an important role in the study of the dynamics of a system because the zero-set {V = 0}, formed by orbits of the system (1), contains the limit cycles and the homoclinic and heteroclinic connections between hyperbolic saddle equilibria which are in Ω [17,20,21,23]. Moreover, the cyclicity of a limit cycle is related to the vanishing order of V [18].
The problem of the existence of an inverse integrating factor depending on the type of singularity has also been considered. Enciso and Peralta-Salas [17] studied the existence of a smooth inverse integrating factor in a neighborhood of an arbitrary elementary singularity, i.e. for the systems whose linear part at the origin have at least one nonzero eigenvalue. This extends previous results given in [11], [14,Theorem 5.2], where the authors consider elementary singularities that admit analytic orbital normalization. For stationary points with nilpotent or vanishing linearization, we only know the results obtained by Walcher [32]. These are partial results and show that generically there is a non-formal inverse integrating factor. Finally, Algaba et al. [7] characterize the nilpotent systems whose lowest-degree quasi-homogeneous term is (y, σx n ) T , σ = ±1, having a formal inverse integrating factor.
In this paper, we study the existence of an algebraic inverse integrating factor (which will be named AIIF) of a nilpotent or degenerate singular point. The class of systems studied here includes, among others, some families of great relevance. For example, the systems (y+· · · )∂ x +(σx n +· · · )∂ y with n a natural number and σ = ±1 and also, the wide class of degenerate systems (−∂h/∂y + · · · )∂ x + (∂h/∂x + · · · )∂ y , with h a homogeneous polynomial of degree 3,4 or 5 having only simple factors on This paper is organized as follows. Section 2 is devoted to provide an expansion in quasi-homogeneous terms of an orbital equivalent normal form of systems whose lowest-degree quasi-homogeneous term is a Hamiltonian vector field, Theorem 2.2.
Our results are presented in Section 3. We obtain in Theorem 3.1 a reduced normal form for the considered class of non-integrable systems. Theorem 3.2 characterizes, by means of normal form theory, when a system of this class admits an AIIF. Moreover, if the vector field has an AIIF, we give the expression of the inverse integrating factor (formal or algebraic) and solve the formal integrability problem. Theorem 3.3 establishes when the origin of such systems is a monodromic point. Theorem 3.4 determines, assuming the existence of an AIIF and the monodromy of the origin, when the origin is either a center or a focus.
In Section 4, we compute the systems with an AIIF for perturbations of quadratic Hamiltonian systems, nilpotent Hamiltonian systems, y 2 ∂ x +x 3 ∂ y and −y 3 ∂ x +x 3 ∂ y . Furthermore, when the origin is a monodromic singular point, we characterize their centers. So, for example, we prove that the system (y 2 + 3 4 ax 3 )∂ x +(x 3 +ax 2 y)∂ y , a = 0, has the AIIF V = (4y 3 − 3x 4 ) 13/12 , but it is not formally integrable and does not have any formal inverse integrating factor (see Proposition 4.7).
Finally, Section 5 contains the proofs of the results obtained.
2. Quasi-homogeneous normal forms. Given t = (t 1 , t 2 ) non-null with t 1 and t 2 non-negative integer numbers without common factors, in what follows, we denote the vector spaces of quasi-homogeneous polynomials and vector fields of type t and degree k by P t k and Q t k , respectively, i.e.
Any vector field can be expanded into quasi-homogeneous terms of type t of successive degrees. Thus, the vector field F can be written in the form for some r ∈ Z, where F j = (P j+t1 , Q j+t2 ) T ∈ Q t j and F r ≡ 0. Such expansions are expressed as F = F r + q-h.h.o.t., where "q-h.h.o.t." means "quasi-homogeneous higher order terms".
If we select the type t = (1, 1), we are using in fact the Taylor expansion, but in general, each term in the above expansion involves monomials with different degrees.
In this paper we will use the following result.
The key in the problem of obtaining a normal form of system (1) is to analyze the effect of a near-identity transformation x = y + P k (y) and a reparametrization of the time by dt dT = 1 + τ k (x), where P k ∈ Q t k and τ k ∈ P t k , with k ≥ 1. The quasi-homogeneous terms of the transformed system y = dy dT = G(y) agree with the original ones up to degree r + k − 1 and for the degree r + k it holds where we have introduced the homological operator under orbital equivalence where [P k , F r ] := DP k F r − DF r P k is the Lie bracket of P k and F r . Following the ideas of the conventional normal form theory, it is enough to choose (P k , τ k ) ∈ Q t k × P t k adequately in order to simplify the (r + k)-degree quasi-homogeneous term in system (1), by annihilating the part belonging to the range of the linear operator L r+k . In other words, we can achieve that F r+k belongs to a complementary subspace to the range of L r+k . When this has been done, we say that the corresponding term has been reduced to normal form under orbital equivalence. So, by means of a sequence of time-reparametrizations and near-identity transformations (by performing the procedure first for k = 1, then for k = 2 and so on) the system (1) can be formally reduced to normal form under orbital equivalence, i.e. the system can be transformed into with G r ≡ 0 and G r+k ∈ Cor(L r+k ) ⊆ Q t r+k where Cor(L r+k ) is any complementary subspace to the range of the homological operator L r+k . We note that such space is not unique, in general.

2.1.
Orbital equivalent normal forms for perturbations of a quasi-homogeneous Hamiltonian system. For a type t fixed, we consider the systems whose quasi-homogeneous expansion iṡ where X h := (−∂h/∂y, ∂h/∂x) T ∈ Q t r , with r a non-negative integer (thus h ∈ P t r+|t| ); i.e. a class of systems which can be considered as perturbations of a Hamiltonian system whose Hamiltonian function h is a quasi-homogeneous polynomial.
For each j > r, we define the linear operator (Poisson bracket of h and µ j−r ) and denote a complementary subspace to the range of the linear operator j (co-range of the operator j ) by Cor( j ). It is always possible to choose the subspaces Cor( j ) such that Cor( r+|t|+j ) = hCor( j ) for all j > r with P t j−r = {0} [3, Proposition 3.18]. We define the following subsets of N 0 : From Lemma 2.1, there exists m 0 := max{N 0 \ I t }. Thus, j / ∈ J 2 , for all j ≥ n 0 := 1 + r + m 0 .
An orbital equivalent normal form for perturbations of a quasi-homogeneous Hamiltonian system is provided in the following result.
Thus, to obtain a normal form, it is enough only to calculate the co-ranges of j for j = r + 1, . . . , n 0 + r + |t| − 1.
We emphasize that if the factorization of h on C[x, y] has only simple factors, then there is, at most, a finite number of monomials g j in the expression of (3).
Here, in this paper, we study the systems satisfying g j ≡ 0, i.e. systems orbitally equivalent to systems of the formẋ = X h + µD 0 .
Such class of systems is a wide family and contains, among others, the nondegenerate saddle (h = xy), linear center (h = x 2 + y 2 ), the nilpotent systems of the form (ẋ,ẏ) = (y, , the systems (−∂h/∂y + · · · )∂ x + (∂h/∂x + · · · )∂ y with h a homogeneous polynomial of degree 3, 4 or 5 and h having only simple factors on C[x, y]. Nevertheless, the system (2) for h = x 6 /6 + y 4 /4 is not orbital equivalent tȯ 3. Main results. We introduce the systems under consideration in this paper. We define F t r+|t| as the set of h ∈ P t r+|t| whose factorization on C[x, y] has only simple factors and satisfies that J is an empty set, i.e. hP t j is a complementary subspace to the range of r+|t|+j for j ≤ r or j such that P t j−r = {0}. So, under Theorem 2.2, the systems we consider are orbitally equivalent to the system (3) with g ≡ 0.
In this section, we characterize the systems (2) with h ∈ F t r+|t| which have an AIIF and we also determine which ones are centers. To achieve our goal, we need to provide a reduced normal form under orbital equivalence of the normal form given by Theorem 2.2.
Let h ∈ P t r+|t| be a polynomial whose factorization on C[x, y] only has simple factors. We consider the systeṁ with µ r+N ≡ 0 and µ j ∈ Cor( j ), j ≥ r + N. The system (4) is a normal form for h ∈ F t r+|t| . We define the linear operator It is always possible to choose Cor( j ) and Cor( j ), co-range of the linear operators j and (2) j , respectively, such that Cor( . Therefore, the normal form provided in the following result is simpler than the normal form (3).  (4)). The following statements are satisfied: 1. if N = j(r + |t|) with j any natural number, then a formal normal form under orbital equivalence for the system (4) iṡ j ), 2. if N = s(r + |t|) with s a natural number, then a formal normal form under orbital equivalence for the system (4) iṡ j ), j = r + 2N. The main result of this paper is stated in the next theorem.
r+|t| possesses an AIIF if, and only if, it is formally orbital equivalent to one, and only one, of the following systems: with N = j(r+|t|) for all natural j and µ r+N ∈ Cor( r+N )\{0} (non-formally integrable system). Moreover, the AIIF is (h + q-h.h.o.t.) 1+N/(r+|t|) , up to a multiplicative constant; i.e. it is not formal.
where N = s(r + |t|) with s a natural number, µ r+N ∈ Cor( r+N ) \ {0} (non-formally integrable system) and α a real number. Moreover, the AIIF is 1+s , up to a multiplicative constant; i.e. it is formal.
We solve the monodromic and the center problems of the system (2). For the monodromy problem, it has the following result. Note that if the origin is a monodromic point and the system is formally integrable, then the origin is a center. Last on, we state the result which gives title to this work. It characterizes the centers of the non-formally integrable systems (2) having an AIIF.
r+|t| and h vanishes only at the origin. Assume that it is a non-formally integrable and possesses an AIIF. Then, the origin is: 1. a center, if I = 0, 2. an unstable focus, if sig(h)I > 0, 3. a stable focus, if sig(h)I < 0, where I = h=sig(h) µ r+N and µ r+N is given by the normal form (4).

Some examples and applications.
In this section we determine several families of the systems (2) admitting an AIIF. For the monodromic case, we also characterize their centers.
The easiest case appears when the polynomial h is quadratic, that is, the lowestdegree part of the system is linear, i.e. systems (2) with t = (1, 1), r = 0 and h = (σx 2 − y 2 )/2 (after a change of variable if it would be necessary) with σ = 1 or −1. The origin of the system (2) is a weak saddle (for σ = 1) or a non-degenerate center-focus (for σ = −1). It is easy to check that h ∈ F (1,1) 2 . One has also that Cor( 2k ) = span{h k } and Cor( j ) = {0}, otherwise. Thus, from Theorem 2.2, these systems are formally orbital equivalent to From Theorem 3.1, a reduced normal form of this normal form is for a certain s, which is the normal form (7) given in Theorem 3.2. Therefore, both have a formal inverse integrating factor. Moreover, they are integrable systems if, and only if, α 2s+1 = α 4s+1 = 0, i.e. they are linearizable. Some related partial results can be seen in [8,11,31].
A) Perturbations of Hamiltonian quadratic systems. These systems are That is, systems (2) with t = (1, 1) and r = 1. From Theorem 3.3, the origin of these systems is non-monodromic. We are looking for systems (8) with an AIIF. For d = 0, without loss of generality, we can assume c = 0 and d = 1, the polynomial h has only simple factors if 27a 2 + 4b 3 = 0, and by Lemma 2.1, the sets P t j are nontrivial spaces for all j. Table 1 shows the range and co-range of the operator j , j = 2, 3, 4 for the system (8) with d = 0. It is straightforward to check that J is an empty set and therefore the polynomial h belongs to F Table 1. Range and co-range of operator j for the system (8).
The normal form (3) of the system (8) becomes Applying Theorem 3.2, we get the following result.
with α a real number and (α 3j+1 , β 3j+1 ) = (0, 0), j ≥ 1. Moreover, the AIIF of the system (8) B) Centers of perturbations of nilpotent Hamiltonian systems. We consider the nilpotent systems whose quasi-homogeneous expansion is of the form From Theorem 3.3, the origin is not monodromic if, and only if, n is even, or n is odd and σ = 1.
We analyze the center problem for the system (9) admitting an AIIF. We assume that the origin is monodromic, i.e. n odd (n = 2m − 1) and σ = −1. These systems are They correspond to systems (2) with t = (1, m), r = m−1 and h = 1 The center problem for the nilpotent systems was solved theoretically in [9,27,30].
We give the following result which characterizes the centers of the nilpotent systems (10) having an AIIF.
Theorem 4.2. Assume that system (10) possesses an AIIF. Then, the origin is a center if, and only if, it is formally orbital equivalent to with B = 0 if 2l = m.
Proof. From [5, Theorem 2], if the system (10) possesses an AIIF then it is formally orbital equivalent either to (ẋ,ẏ) T = (y, −x 2m−1 ) T which is a center, or to with A a nonzero real number, f a function with f (0) = 1, L a non-negative integer, By applying Theorem 3.2, we obtain a further reduction of the normal form (12) of the system (10): In order to get the centers, it is enough to compute the integral I given by Theorem 3.4. In this case I = A It is known that the integral I is different from zero if, and only if, M is even. So, M = 2l − 1; that is, the system (13) becomes the system (11).
The sufficient condition follows since the system (11) is time-reversible and its origin is a monodromic singular point.
We note that if 2l = m and A = 0, the inverse integrating factor is not formal. Consequently, the centers of the systems (10) having an AIIF are formally orbitalequivalent to time-reversible systems but not all of them have a formal inverse integrating factor. C) Perturbations of the system y 2 ∂ x + x 3 ∂ y . We consider the degenerate systems of the form with P j and Q j homogeneous polynomials of degree j and Q 3 (1, 0) = 0 (without loss of generality, we can assume Q 3 (1, 0) = 1). The quasi-homogeneous expansion with respect to t = (3, 4) of the system (14) is  Table 2. Range and co-range of operator j for the system (15) Range Range( 10 )=span{0}, Cor( 10 )=span{x 2 y} As above, we observe that h belongs to F (3,4) 12 . So, from Theorem 2.2, a normal form of the system (15) becomes with f j ∈ span{x 2 , xy, x 2 y, h, xh, yh}.
We claim that the AIIF's of the nonformally integrable systems (14) are algebraic but not formal. Consequently, we get the following result.  (14) is formally integrable if, and only if, it admits a formal inverse integrating factor.
Next, we give necessary conditions for the existence of an AIIF for the system (14). The first two coefficients of the right-hand side of (16) are We have obtained these coefficients of the quasi-homogeneous normal form by using the procedure given in [1]. From Theorem 4.3, we have the following result.

D) Perturbations of the system −y
i.e. the system (2) with t = (1, 1) and r = 2. From Lemma 2.1, the sets P t j are nontrivial spaces for all j, hence n 0 = 1 + r = 3. So, to provide a normal form, it is enough to compute the subspaces Cor( j ), j = 3, 4, 5, 6, which are given in Table  3. Table 3. Range and co-range of operator j for the system (21).
Other properties of these integrals can be found in [22].
Applying Theorems 3.2 and 3.4, we have the following result. 1. (ẋ,ẏ) T = X h . The AIIF is g(h) with g any nonzero function (in particular, there are nonzero inverse integrating factors at the origin).In this case, the origin is a center.
The AIIF is (h + q-h.h.o.t.) 2+j+1/4 , up to a multiplicative constant. In this case, the origin is a center.
The AIIF is (h + q-h.h.o.t.) 2+j+3/4 , up to a multiplicative constant. In this case, the origin is a center. Example 4. We analyze the system This system for c 3 = 1/2 and c 4 = 0 was studied in Moussu [27] by showing that it is a degenerate analytic center without formal first integral. Moreover, Giné and Peralta-Salas [24] have proved that it does not have a formal inverse integrating factor. The first coefficients of the normal form (22) are α 3 = 2c 3 and β 3 = 3c 4 . If both c 3 and c 4 are zero, the system is Hamiltonian whose first integral is h = x 4 + y 4 . Otherwise, the system is not formally integrable. Moreover, • if c 3 .c 4 = 0, the fourth-order coefficients of the normal form are α 4 = 0 and β 4 = 2c 3 c 4 . Thus, from Theorem 4.9, the system does not have an AIIF, • if c 3 = 0 and c 4 = 0, the fourth-order coefficients of the normal form are zero but α 5 = 6/5c 3 4 . And if c 3 = 0 and c 4 = 0, then β 5 = 16/45c 3 3 . Therefore, from Theorem 4.9, the system does not have an AIIF.
These results are summarized below.

5.
Proofs of the main results. We consider the non-integrable system (4).
It is possible to compute a reduced normal form by proceeding as follows.
We first assume that r = 0 and we distinguish two cases: We suppose that l 2 = 0. As Ker( r+k−N ) is a trivial space, then Therefore, Range(L (2) r+k ) = Range(L r+k ). For both cases, l 2 = r or l 2 = r, r+k )D 0 .
We now suppose that l 2 = 0. We have that Ker with α a real number. Consequently, if there exists a natural number l 1 := s such that N = s(r + |t|), i.e. r + k = r + 2s(r + |t|), then Range(L (2) r+2s(r+|t|) ) = Range(L r+2s(r+|t|) ) and Cor(L and particularizing in our context, if a system has an AIIF then it also admits an inverse integrating factor V over C((x, y)) where C((x, y)) denotes the quotient field of the algebra of the power series C[[x, y]]. Moreover, if V exists, from Algaba et al. [5], then the system also has an inverse integrating factor of the form W q with W ∈ C[[x, y]] and q a rational number.
The following results are used for the proof of Theorem 3.2.
Proposition 5.5. [7,Proposition 19] Consider the systemẋ = X h + µD 0 with µ = j>r µ j , µ j ∈ Cor( j ) and h ∈ P t r+|t| having only simple factors in its factorization on C[x, y] and µ ≡ 0. We assume that V is an inverse integrating factor V of the system. Then, V = h m+1 + j>m+1 b j h j , for a certain m natural number.
Moreover, it holds with V j the quasi-homogeneous term of degree j of V.
Proposition 5.6. Consider the systemẋ = X h + λg(h)D 0 where h ∈ P r+|t| , λ ∈ P r+N and g a C 1 class function with g(0) = 1. Then, the function h 1+N/(r+|t|) g(h) is an inverse integrating factor of the system.
Proof. Applying the Euler theorem for quasi-homogeneous functions, one has that  (2) is of the formẋ = X h + µD 0 , with µ = j>r µ j , µ j ∈ Cor( j ). If µ j ≡ 0, for all j then the system (2) is formally orbital-equivalent to a Hamiltonian system, i.e. there exist a diffeomorphism Φ and a function η on U ⊂ C 2 with detDΦ = 0 on U and η(0) = 0, such that Φ * (ηF) = X h , where we have denoted as Φ * the push-forward defined by Φ. As X h is a Hamiltonian vector field, f (h) is a first integral for any f non-constant. In particular, f (h) is an inverse integrating factor. So, the pull-back Φ * brings f (h) on the inverse integrating factor of the system (2), V=f (h + · · · ) + · · · , i.e. it is not unique. Also, if f (0) = 0, V would be a formal inverse integrating factor with V (0) = 0.
Otherwise, let N = min{j, µ r+j = 0}. We distinguish two cases: • N is not multiple of r + |t|.
Therefore, m = s. Thus, up to a multiplicative constant the leading term in the quasihomogeneous expansion of any formal inverse integrating factor is h s+1 .
For k = j 0 − r + (s + 1)(r + |t|), the equality (28) has two factors V (s+1)(r+|t|)μj0 and V (2s+1)(r+|t|)μj0−s(r+|t|) . This second term is zero. Thus, Equation (28)  and, as V (s+1)(r+|t|) = h s+1 and j 0 −r = s(r+|t|), we obtain µ j0 = 0, a contradiction. Therefore, the existence of a formal inverse integrating factor implies that a normal form under orbital-equivalence of the system (2) is the system (7). Sufficient condition. We suppose that the system (2) is orbital equivalent toẋ = X h . Any function f (h) is a first integral and, in particular, it is an inverse integrating factor. Thus, if we perform the transformation which bringsẋ = X h to the system (2), we have that the system (2) admits an AIIF but it is not unique.
Otherwise, let N = min{j, µ r+j = 0}. The normal forms (6) and (7) are of the formẋ = X h + λg(h)D 0 , with λ = µ r+N ∈ P r+N and g(0) = 1. From Proposition 5.6, the function h 1+ N r+|t| g(h) is an AIIF of the system (6) and (7)  Proof of Theorem 3.3. Assume that the factorization of h ∈ P t r+|t| only has simple factors. Therefore, we can write in a compact form h = c n j=1 f j m j=1 g j , where f j = x, y or y t1 −λ j x t2 , j = 1, . . . , n, g j (x, y) = (y t1 −a j x t2 ) 2 +b 2 j x 2t2 , j = 1, . . . , m with c, λ j , a j and b j real numbers and λ j , b j nonzero, for all j.
We shall prove the necessary condition. We assume the contrary one. Thus, there exists a f j and it is a simple real factor of h. Therefore, there exists a real orbit of the system which leaves or enters at the origin [6,Proposition 8]. Consequently, the origin is not monodromic.
We see the sufficient condition. If h is different from zero for any (x, y) = (0, 0), one has that the origin is monodromic [6,Proposition 6].
Proof of Theorem 3.4. As the system (2) is non-integrable and possesses an AIIF, its reduced normal form is either (6) or (7). First, we assume that the system (2) is formally orbital-equivalent to the system (6).
We also can assume that h(x, y) is positive for all (x, y) = (0, 0) since, from Theorem 3.3, if the origin is monodromic, h preserves its sign and, if h is negative by changing the time t by −t, h becomes −h.
Finally, we make the change z = − 1 N u −N , that converts system (33) into z = µ r+N (θ), A Poincaré map for the system (34) is Π(z 0 ) = z(T, z 0 ) = z 0 + I. Consequently, the result follows for N not multiple of r + |t|.
Following the same reasoning as before, we complete the proof.