On the integrability problem for the Hopf-zero singularity and its relation with the inverse Jacobi multiplier

arXiv:1909.04398v1 [math.DS] 10 Sep 2019 On the Integrability Problem for the Hopf-Zero singularity and its relation with the inverse Jacobi multiplier A. Algaba†, N. Fuentes†, E. Gamero‡, C. Garc´ıa† †CEAFMC. Faculty of Experimental Sciences University of Huelva, Spain ‡Dept. Applied Mathematics II, ETSI. University of Sevilla, Spain September 11, 2019 Abstract In this paper we use the orbital normal form of the nondegenerate Hopf-zero singularity to obtain necessary conditions for the existence of first integrals for such singularity. Also, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. Some algorithmic procedures for determining the existence of first integrals are presented, and they are applied to some families of vector fields. 1 Introduction Let us consider an analytic three-dimensional system that undergoes a linear degeneracy corresponding to a zero and a pair of pure imaginary eigenvalues. By translating the equilibrium point to the origin and using a linear transformation, the Hopf-zero singularity can be written as   ˙x ˙y ˙z  =  −y x 0  +  f(x, y, z) g(x, y, z) h(x, y, z)  , where f, g, h are analytic functions at the origin that denote the nonlinear terms. We consider the nondegenerate Hopf-zero singularity, that arises by assuming the generic conditions ∂2h ∂x26= 0 or ∂2h ∂y26= 0. Under this hypothesis, it is a simple matter to show that the above system can be expanded in quasi-homogeneous terms of type t= (1,1,2):   ˙x ˙y ˙z  =  −2y 2x x2+y2 +  F1(x, y, z) G1(x, y, z) H1(x, y, z)  +  F2(x, y, z) G2(x, y, z) H2(x, y, z)  +··· ,(1.1) 1 where Fk= (Fk(x, y, z), Gk(x, y, z), Hk(x, y, z))Tis a quasi-homogeneous vector field of type t and degree k. We notice that the principal part (the lowest-degree quasi-homogeneous term, which has degree 0) can be expressed as F0(x) =   −2y 2x x2+y2 =Xh h,(1.2) where x= (x, y, z) and Xh=−∂h ∂y,∂h ∂xT = (−2y, 2x)T(1.3) denotes the planar Hamiltonian vector field with Hamiltonian function h=x2+y2. In this paper, we study the existence of first integrals in a neighborhood of the equilibrium point at the origin for this kind of systems (recall that a first integral is a non-constant function that is constant when it is evaluated along any solution of the system). Revealing the existence of first integrals for a given system is very useful to understand its qualitative behavior. Namely, for a planar system the existence of a first integral determines completely its phase portrait. For higher-dimensional systems, this can be done by obtaining a sufficient number of functionally-independent first integrals. The three-dimensional center problem for the Hopf-zero singularity (that consists of determining whether there is a neigbourhod of the singularity foliated by periodic orbits, including a curve of equilibria) has been analyzed in [17, 18]. This is equivalent to the integrability problem, that consists of determining the existence of a pair of functionally-independent first integrals. In the quoted works, it is shown that a Hopf-zero singularity is completely integrable if, and only if, it is orbitally equivalent to its linear part (−y, x, 0)T. Moreover, in the case of integrability, there are two functionally-independent first integrals of the form I1=h+··· and I2=z+··· (the dots denote higher-order terms). It is a simple matter to show that, in the nondegenerate Hopf-zero singularity (1.1) that we are considering, there are no first integrals of the form I2=z+···. In other words, the nondegenerate Hopf-zero singularity (1.1) can not be completely integrable and our analysis will be focused on detecting the existence of first integrals for such a singularity of the form I=h+···. This is still a difficult problem and there are few known satisfactory methods to solve it. In the present paper, we use the orbital normal form for system (1.1) obtained in [3] to establish necessary conditions for the existence of first integrals for the nondegenerate Hopf-zero singularity (1.1). Moreover, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. In the planar nilpotent case, analogous relations concerning first integrals and inverse integrating factors has been obtained in [4, 6, 9, 10]. This paper is organized as follows. In Section 2 we include definitions and results about quasi-homogeneous vector fields and the nondegenerate Hopf-zero orbital normal form, that we use in this work (their proofs can be found in [3]). In Section 3, the orbital normal form is used to obtain some results about the analytic integrability of this singularity. The main result is Theorem 3.6, that determines the existence of an analytical first integral for the nondegenerate 2 Hopf-zero singularity in terms of the vanishing of some normal form coefficients. Moreover, an algorithmic procedure for obtaining necessary conditions for the existence of first integrals of polynomial vector fields, that is applicable under some hypothesis in the orbital normal form, is presented. Section 4 analyzes the relation between the existence of first integrals and the existence of inverse Jacobi multipliers for the nondegenerate Hopf-zero singularity. In particular, based on an algorithm to determine the existence of inverse Jacobi multipliers, we present a new algorithmic procedure to determine the existence of first integrals of polynomial vector fields, that is applicable in all the cases. Finally, in Section 5 we apply the results to a couple of three-parameter families of vector fields, where we find all the cases of existence of analytical first integrals. 2 The Hopf-zero orbital normal form In this section, we collect results from [3] that will be used along this paper. Among them, the main result is Theorem 2.2, where we present the orbital normal form for system (1.1). Before we state it, we introduce some definitions and results. We say that a scalar function fof nvariables is quasi-homogeneous of type t= (t1,...,tn)∈ Nnand degree kif f(ǫt1x1,...,ǫtnxn) = ǫkf(x1,...,xn). A vector field F= (F1,...,Fn)Tis quasi-homogeneous of type tand degree kif Fj∈Pt k+tjfor j= 1,...,n. The vector spaces of quasi-homogeneous functions and vector fields of type tand degree kare denoted, respectively, by Pt kand Qt k. In this paper, we use the type t= (1,1,2) (for functions and vector fields depending on three variables), as well as the type ˆ t= (1,1) (, that appear when dealing with functions and vector fields depending on two variables. For instance, we have F0∈ Qt 0,Xh∈ Qˆ t 0and h∈Pˆ t 2. A conservative-dissipative decomposition of quasi-homogeneous planar vector fields has been used in [5] in the study of the integrability problem. In the specific case that we are considering, this decomposition reads as follows. Proposition 2.1 Let us consider Pk∈ Qˆ t kand denote D0= (x, y)T∈ Qˆ t 0. Then, there exist unique quasi-homogeneous polynomials hk+2 ∈Pˆ t k+2 and νk∈Pˆ t ksuch that: Pk=Xhk+2 +νkD0.(2.4) Moreover, hk+2 =1 k+2D0∧Pkand νk=1 k+2div(Pk). In the above proposition, we have introduced the wedge product of two planar vector fields F= (P, Q)T,G= (R, S)T, defined by F∧G=P S −Q R (see [19]) and the divergence div (F) = ∂P ∂x +∂Q ∂y . We notice that, if F∈ Qˆ t kand G∈ Qˆ t l, then F∧G∈Pˆ t k+l+2 and div (F)∈Pˆ t k. Next, we present the orbital normal form for system (1.1) obtained in [3]. It determines how much system (1.1) can be simplified by means of an infinite sequence of time-reparametrizations and near-identity coordinate transformations. In fact, the orbital normal form presented is formal, which indicates that we will not discuss matters of convergence. 3 Theorem 2.2 A formal normal form under orbital equivalence for system (1.1) is ˙ x=G(x) = F0(x) + G1(z)D0 G2(z),(2.5) where G1(z) = P∞ k≥1akzkand G2(z) = P∞ k≥1bkzk+1. Observe that the 0th-degree quasi-homogeneous term of the orbital normal form (2.5) agrees with those of system (1.1). Moreover, the kth-degree quasi-homogeneous term of the orbital normal form (2.5) is Gk(x) = akzkD0 bkzk+1 ∈ Qt k. In the rest of this paper we use the orbital normal form (2.5) to study the integrability problem for the nondegenerate Hopf-zero singularity. The first nonzero term in the Taylor expansions of functions G1and G2play an outstanding role in this analysis. Therefore, let us denote l0:= min {l∈N:al6= 0}, m0:= min {m∈N:bm6= 0}.(2.6) We notice that G1(u)≡0 if, and only if, l0= +∞, and G2(u)≡0 if, and only if, m0= +∞. 3 The integrability problem for the Hopf-zero singularity In this subsection, we show that the orbital normal form (2.5) is useful in the analysis of the integrability problem (consisting of determining the existence of a first integral) for the nondegenerate Hopf-zero singularity (1.1). Recall that a function Iis called a first integral for system (1.1) if Iis constant when it is evaluated along any solution of the system. If Iis a C1function, using the chain rule, this means that ∇I·F= 0. Our first result states that the analysis of the integrability problem for analytic systems can be reduced to the formal context through formal diffeomorphisms. Proposition 3.3 Let us consider the system ˙ x=F(x), where Fis an analytic vector field and the transformation x= Φ(e x), where Φis a formal diffeomorphism. Then, system ˙ x=F(x) admits an analytical first integral if, and only if, the transformed system ˙ e x=e F(e x)admits a formal first integral. Proof: The necessary condition is trivial, because if Iis an analytical first integral for system ˙ x=F(x), then e I:= I◦Φ is a formal first integral for system ˙ e x=e F(e x). To prove the sufficient condition, let us denote by e Ia formal first integral of system ˙ e x=e F(e x). Then, ˆ I=e I◦Φ−1is a formal first integral of system ˙ x=F(x). From Theorem A of [20], there 4 exists a formal scalar function ˆ lsuch that ˆ l(0) = 0, ˆ l′(0) = 1, such that I=ˆ l◦ˆ Iis an analytical first integral for system ˙ x=F(x). We notice that the orbital normal form (2.5), as well as those obtained in [15, 16], is invariant under rotations. Hence, the first integrals depend on x2+y2and z. Next result uses the orbital normal form (2.5) to determine the existence of a formal first integral for the nondegenerate Hopf-zero singularity by reducing it to a nilpotent singularity. If we use instead the orbital normal form given in [15, 16], then an integrability problem for a planar system with null linear part arises, which is more difficult to solve. Proposition 3.4 The orbital normal form (2.5) admits a formal first integral if, and only if, the planar system ˙u=v+G2(u), ˙v= 2vG1(u),(3.7) is formally integrable. Proof: Let us consider cylindrical coordinates x=ρsin(θ), y=ρcos(θ), z=u, and the singular change v=ρ2. Then, the normal form (2.5) becomes: ˙u=v+G2(u), ˙v= 2vG1(u), ˙ θ= 2. It is enough to remove the azimuthal component to complete the proof. We denote the vector field corresponding to planar system (3.7) by P(u, v) = v+G2(u) 2vG1(u).(3.8) Next result provides a necessary condition for the existence of analytical first integrals for system (1.1), which determines the structure of the quasi-homogeneous normal form in case of existence of analytical first integrals. Proposition 3.5 Let us assume that system (1.1) admits an analytical first integral. Then, its formal orbital normal form (2.5) is given in one of the following items: (a) ˙ x=F0(x). (b) ˙ x=F0(x) + Fs(x) + ···, where s∈Nand Fs∈ Qt sis one of the following vector fields: (b.1) Fs(x) = al0zl0D0 0∈ Qt s, where s= 2l0and al0∈R\ {0}. (b.2) Fs(x) = 0 bm0zm0+1∈ Qt s, where s= 2m0and bm0∈R\ {0}. 5 (b.3) Fs(x) = am0zm0D0 bm0zm0+1 ∈ Qt s, where s= 2m0and am0, bm0∈R\ {0}satisfy 2n1am0+ (m0+ 1)n2bm0= 0, for some n1, n2∈Ncoprime (i.e., their greatest common divisor is 1). Proof: From Proposition 3.3, we obtain that system (1.1) admits an analytic first integral if, and only if, its orbital normal form (2.5) has a formal first integral. From Proposition 3.4, this occurs if, and only if, the planar system (3.7) is formally integrable. In this case, if we select an arbitrary type ˆ t∈N2, the principal part of the vector field Pgiven in (3.8) must be polynomially integrable. Let us consider the following situations: •Case G1(u)≡G2(u)≡0. Let us take the type ˆ t= (1,1). Then, the principal part of P is P0(u, v) = (v, 0) and ˆ I=vis an analytic first integral. This case is considered in item (a) of the statement. •Case G1(u)6≡ 0 or G2(u)6≡ 0. Let us denote n0:= min {n∈N: 2an+ (n+ 1)bn6= 0}.(3.9) We notice that min {m0, l0} ≤ n0and it is possible that l0= +∞,m0= +∞or n0= +∞, but the situation l0=m0= +∞can not occur. The following sub-cases can arise: –If l0< m0, taking the type ˆ t= (1, l0+ 1), the principal part of Pis Pl0(u, v) = (v, 2al0ul0v)T∈ Qˆ t l0, and ˆ I=v−2al0 l0+1ul0+1 is an analytic first integral of Pl0. This case corresponds to item (b.1). –If m0< l0, taking the type ˆ t= (1, m0+ 1), the principal part of Pis Pm0(u, v) = (v+bm0um0+1,0)T∈ Qˆ t m0, and ˆ I=vis an analytic first integral of Pm0. This is the case (b.2). –If m0=l0< n0, then we have 2am0+(m0+1)bm0= 0. Taking the type ˆ t= (1, m0+1), the principal part of Pis Pm0(u, v) = (v+bm0um0+1,−(m0+ 1)bm0um0v)T∈ Qˆ t m0, which is a Hamiltonian vector field, with Hamiltonian function −bm0um0+1v−1 2v2. Therefore, Pm0is polynomially integrable. This case is presented in item (b.3) with n1=n2= 1. –If m0=l0=n0<+∞, then we have 2am0+ (m0+ 1)bm06= 0. Taking the type ˆ t= (1, m0+ 1), the principal part of Pis Pm0(u, v) = (v+bm0um0+1,2am0um0v)T∈ Qˆ t m0, 6 which is integrable because Palso is. Let us consider the conservative-dissipative splitting (2.4) for Pm0. The Hamiltonian function in the quoted splitting is h=−1 2vv−2am0−(m0+1)bm0 2(m0+1) um0+1. Then, S1≡v= 0,and S2≡v−2am0−(m0+1)bm0 2(m0+1) um0+1 = 0, are invariant curves of Pm0, with cofactors K1= 2am0um0and K2= (m0+1)bm0um0, respectively. Since Pm0is polynomially integrable, there exist n1, n2∈Ncoprime such that Sn1 1Sn2 2is a polynomial first integral of Pm0, i.e. n1K1+n2K2= (2n1am0+n2(m0+ 1)bm0)um0= 0. This case is presented also in item (b.3). Next, we present a necessary and sufficient condition for the existence of an analytic first integral of system (1.1). Theorem 3.6 System (1.1) admits an analytical first integral if, and only if, its formal orbital normal form (2.5) is given in one of the following items: (a) ˙ x=F0(x). In this case, there exists a first integral of the form I=h+···. Moreover, there is a curve of equilibria passing through the origin surrounded by an infinity of invariant cylinders. (b.1) ˙ x=F0(x) + G1(z)D0 0, with G1(z) = Pk≥1akzk. In this case, there exists a first integral of the form I=h+···. Moreover, there is a curve of equilibria passing through the origin. (b.2) ˙ x=F0(x) + 0 G2(z), with G2(z) = Pk≥1bkzk+1. In this case, there exists a first integral of the form I=h+···. (b.3) ˙ x=F0(x) + Fm0(x), where Fm0(x) = am0zm0D0 bm0zm0+1 ,m0∈N, and am0, bm0∈R\ {0} satisfy 2n1am0+(m0+1)n2bm0= 0 for some coprime natural numbers n1, n2. In this case, there exists a first integral of the form I=hn1+n2+···. In the above expressions, the dots denote higher-order quasi-homogeneous terms. Proof: Firstly, we prove the sufficient condition. (a) The normal form ˙ x=F0(x) has the first integral e I=h. Undoing the normalizing transformations, we obtain that I=h+··· is a first integral for system (1.1). 7 (b.1) The normal form ˙ x=F0(x)+G1(z)D0 0admits the first integral e I=h−2Rz 0G1(ξ)dξ. Undoing the normalizing transformations, we obtain that I=h+··· is a first integral for system (1.1). (b.2) The normal form ˙ x=F0(x) + 0 G2(z)has the first integral e I=h. Undoing the normalizing transformations, we obtain that I=h+··· is a first integral for system (1.1). (b.3) If n1=n2= 1 (Hamiltonian case), then e I=h2+ 2bm0zm0+1his a first integral for the normal form ˙ x=F0(x) + Fm0(x). Undoing the normalizing transformations, we obtain that I=h2+··· is a first integral for system (1.1). Otherwise (dissipative case), the normal form ˙ x=F0(x) + Fm0(x) admits the first integral e I=hn1h−2am0−(m0+1)bm0 2(m0+1) zm0+1n2. Undoing the normalizing transformations, we obtain that I=hn1+n2+··· is a first integral for system (1.1). Next, we prove the necessary condition. Let us assume that system (1.1) admits an analytical first integral and consider its orbital normal form (2.5). From Proposition 3.5, the quoted formal orbital normal form is either ˙ x=F0(x) (that corresponds to the item (a) of the statement) or ˙ x=F0(x) + Fs(x) + ···, where Fs∈ Qt sis given in one of the cases of item (b) of Proposition 3.5. We deal with each case separately. (b.1) Here, Fs(x) = al0zl0D0 0∈ Qt s, where s= 2l0, and al0∈R\ {0}. To complete the proof in this case, it is enough to show that G2(z)≡0 in the orbital normal form (2.5). We use reductio ad absurdum: if G2(z)6≡ 0 then m0<+∞. Taking the type ˆ t= (1, l0+1), the principal part of planar system (3.7) is Pl0(u, v) = (v, 2al0ul0v)T∈ Qˆ t l0. Let us denote the formal first integral of system (3.7) by ˆ I. We notice that ˆ G(u, v) = v−2Ru 0G1(ξ)dξ is a first integral of the vector field (v, 2G1(u)v)T. If we define ˆ H=ˆ I−ˆ G, we have: ∇ˆ I·P=∇(ˆ G+ˆ H)·P=∇ˆ G·P+∇ˆ H·P=∇ˆ G·(G2(u),0)T+∇ˆ H·P= 0. In this equality, the quasi-homogeneous term of degree l0+m0+ 1 is given by −2al0bm0ul0+m0+1 +∇ˆ Hm0+1 ·Pl0=−2al0bm0ul0+m0+1 +v∇ˆ Hm0+1 ·(1,2al0ul 0)T= 0, where ˆ Hm0+1 ∈Pˆ t m0+1 is the quasi-homogeneous term of degree (m0+1) of ˆ H. Nevertheless, the above equation is incompatible. Hence, system (3.7) can not admit any first integral and, applying Proposition 3.4, system (1.1) does not admit any formal first integral, which is contradictory. (b.2) Now, Fs(x) = 0 bm0zm0+1∈ Qt s, where s= 2m0, and bm0∈R\ {0}. To complete the proof in this case, it is enough to show that G1(z)≡0 in the orbital normal form (2.5). 8 Again, we use reductio ad absurdum: if G1(z)6≡ 0 then l0<+∞. Taking the type ˆ t= (1, m0+ 1), the principal part of planar system (3.7) is Pm0(u, v) = (v+bm0um0+1,0)T∈ Qˆ t m0. Let us denote the formal first integral of system (3.7) by ˆ I. We notice that ˆ G(u, v) = vis a first integral of the vector field (v+G2(u),0)T. If we define ˆ H=ˆ I−ˆ G, then: ∇ˆ I·P=∇(ˆ G+ˆ H)·P=∇ˆ G·P+∇ˆ H·P=∇ˆ G·(0,2G1(u)v)T+∇ˆ H·P= 0. In this equality, the quasi-homogeneous term of degree l0+m0+ 1 is given by 2al0ul0v+∇ˆ Hm0+1 ·Pm0= 2al0ul0v+∂ˆ Hm0+1 ∂u (v+bm0um0+1) = 0, where ˆ Hm0+1 ∈Pˆ t m0+1 is the quasi-homogeneous term of degree (m0+ 1) of ˆ H. As in the previous subcase, the above equation is incompatible, and system (3.7) can not admit any first integral. Hence, by applying Proposition 3.4, we deduce that system (1.1) does not admit any formal first integral, which is contradictory. (b.3) In this case, we have Fm0(x) = am0zm0D0 bm0zm0+1 ∈ Qt s, where m0∈N, and am0, bm0∈R\{0} satisfy 2n1am0+ (m0+ 1)n2bm0= 0 being n1, n2∈Ncoprimes. There are two cases to be considered. The first one corresponds to the free-divergence case, that arises if n1=n2= 1, and then 2am0+ (m0+ 1)bm0= 0. Taking the type ˆ t= (1, m0+ 1), the principal part of planar system (3.7) is Pm0(u, v) = (v+bm0um0+1,−(m0+ 1)bm0um0v)T∈ Qˆ t m0. This is a Hamiltonian vector field, with Hamiltonian function −bm0um0+1v−1 2v2. Using Corollary 4.23 of [5], we obtain that system (3.7) is integrable if, and only if, it is formally equivalent to ( ˙u, ˙v) = Pm0(u, v). Then, from Proposition 3.4 we obtain that system (1.1) admits a first integral if, and only if, it is formally equivalent to ˙ x=F0(x)+ Fm0(x).This falls into the item (b.3) of the statement. In the second case (non-zero divergence), we have 2n1am0+ (m0+ 1)n2bm0= 0 for some n1, n2∈Ncoprimes with n16= 1 or n26= 1. Taking the type ˆ t= (1, m0+ 1), the principal part of planar system (3.7) is Pm0(u, v) = (v+bm0um0+1,2am0um0v)T∈ Qˆ t m0, which is integrable because vn1v−2am0−(m0+1)bm0 2(m0+1) um0+1n2is a first integral and besides div (Fm0) = 2am0+ (m0+ 1)bm06= 0. Using Theorem 1.2 of [8], we obtain that Pis orbitally equivalent to Pm0. This case corresponds to the item (b.3) of the statement. 9 where Rk= k−1 X j=s (∇Mj·Fk−j−Mjdiv (Fk−j)) ∈Pt k. To complete the proof it is enough to argue as in the proof of Lemma 3.7. The proof of item (b) is analogous. Theorem 4.12 Let us consider system (1.1). (a) Let assume that the formal orbital normal form (2.5) falls into the case (b.3) of Proposition 3.5, and consider the unique scalar function Mintroduced in Lemma 4.11 satisfying (4.13). Then, system (1.1) admits an analytical first integral if, and only if, βk= 0 for all k. (b) Let assume that the formal orbital normal form (2.5) falls into the cases (a),(b.1) or (b.2) of Proposition 3.5, and consider the unique scalar function e Mintroduced in Lemma 4.11 satisfying (4.14). Then, system (1.1) admits an analytical first integral if, and only if, e βk= 0 for all k. Proof: (a) To prove the necessary condition, we observe that if βk= 0 for all k≥3, then Mis an inverse Jacobi multiplier. From Proposition 4.10, system (1.1) admits a formal first integral and finally, from Proposition 3.3 we deduce that system (1.1) admits an analytical first integral. Let us prove the sufficient condition by reductio ad absurdum. Let us suppose on the contrary that there exists a scalar function M=h2+Pk≥3Mk, with Mk∈Pt k, such that ∇M·F−Mdiv (F) = βNzN+···, with βN6= 0. Also, we assume that system (1.1) admits an analytical first integral. From Proposition 3.5, there exist a time-reparametrization µ (satisfying µ(0) = 1) and a near-identity transformation x=φ(e x), bringing system (1.1) into its orbital normal form (2.5), which corresponds to the vector field G=Fs+···, being Fs=am0zm0D0 bm0zm0+1 , where s= 2m0and 2n1am0+ (m0+ 1)n2bm0= 0 for some n1, n2∈Ncoprime. Let us define b M(e x) = µ(e x)M(φ(e x)) det (Dφ(e x)) =h2+··· . We have ∇b M·G−b Mdiv (G) = det (Dφ(e x)) µ2(e x)∇b M·F−b Mdiv (F)=βNzN+··· .(4.15) On the other hand, it is easy to show that ∇b M·G−b Mdiv (G) = zm0am0ˆ ∇b M·D0+bm0z∂b M ∂z −(2am0+ (m0+ 1)bm0)bm0b M. 16 Observe that, as βN6= 0, there is a term of the form CzN−m0in the analytical expression of b M. Then, the term C(N−m0)zN−m0(which has quasi-homogeneous degree 2N−2m0< 2N) appears in the analytical expression of ∇b M·F0because it can not be annihilated by the operator ˆ ℓ2N−2m0(it is not in the range of this linear operator). But this is contradictory because the left hand side of (4.15) has quasi-homogeneous degree 2N−2m0<2Nand the right hand side of has degree greater than 2N. (b) The proof of this item is analogous to the proof of Theorem 3.8. This theorem allows to define an algorithm for obtaining necessary conditions for the integrability of a polynomial vector field. Namely, if its formal orbital normal form (2.5) falls into the case (b.3) of Proposition 3.5, it is enough to look for the unique function of the form M=h2+··· (specified in item (a) of Lemma 4.11) and then discard cases of non-integrability from the conditions β26= 0,... If the orbital normal form (2.5) falls into the cases (a),(b.1) or (b.2) of Proposition 3.5, it is enough to look for the unique inverse Jacobi multiplier of the form e M=h+··· (specified in item (b) of Lemma 4.11) and then discard cases of non-integrability from the conditions e β26= 0,... In this case, we could also apply the ideas presented in the previous subsection to determine the non-integrability cases. 5 Some particular cases In this last section, we consider two three-parameter families of vector fields. The first one corresponds to the family:   ˙x ˙y ˙z  =  −2y 2x x2+y2 +  a001z b200x2 c030y3 .(5.16) Next theorem determines the cases where the above family admits an analytical first integral. Theorem 5.13 System (5.16) admits an analytical first integral if, and only if, a001 = 0. Proof: A simple computation shows that the first coefficients of the formal orbital normal form (2.5) for system (5.16) are: a1=−3a2 001/8, b1= 3 a2 001/8. Then, two situations can arise: (a) If a001 = 0, then system (5.16) admits the first integral I(x, y) = x2+y2+b200x3. Moreover, M=Iis an inverse Jacobi multiplier of the form M=h+···. (b) If a001 6= 0, from Theorem 2.2, we obtain that Fis orbitally equivalent to the vector field F0+F1+···, where F1=a1zD0 b1z2with a1,b1given before. 17 As a16= 0, b16= 0 and a1+b1= 0, from Theorem 4.10 we obtain that system (2.5) is integrable if, and only if, it admits an inverse integrating factor of the form M=h2+···. From Lemma 4.11, this occurs if αi= 0 for all 2i > 4. In this case, we have obtained α6=···=α12 = 0, and α14 =8 3a8 001 126 a2 001 −117 b200a001 + 40 b2 200. The vanishing of α14 implies 126 a2 001 −117 b200a001 + 40 b2 200 = 0. Under this hypothesis, we have obtained α16 =···=α20 = 0, and α22 =a3 001 2560(1814374881 b200 −1620931148 a001)c2 030 +(10742498602234 a001 −6174238921423 b200 )a2 001. Assuming now that α22 is zero, we have also obtained that α24 =α26 =α28 = 0, but α30 6= 0. Hence, system (5.16) is not integrable in this case. The second family that we consider is the following:   ˙x ˙y ˙z  =  −2y 2x x2+y2 +  a001z 0 c101xz +c011yz  .(5.17) Next theorem solves the integrability problem for the above family. Theorem 5.14 Let us assume that system (5.17) admits an analytical first integral. Then, one of the following conditions holds: (i) a001 = 0. (ii) a001 6= 0,a001 +c011 = 0,c101 = 0. (iii) a001 6= 0,a001 + 2c011 = 0. Proof: A simple computation shows that the first coefficients of the normal form (2.5) for system (5.17) are: a1=−3a001(a001 +c011)/8, b1= 3a001(a001 + 2c011)/8. We consider the following situations: (a) If a001 = 0, then system (5.17) admits the first integral I(x, y) = x2+y2. Moreover, M=I is an inverse Jacobi multiplier of the form M=h+···. This situation corresponds to item (i) of the statement. (b) If a001 6= 0, from Theorem 2.2 we obtain that the vector field Fis orbitally equivalente to F0+F1+···, where F1=a1zD0 b1z2and a1,b1are given before. The following sub-cases can arise: 18 (b1) If a001 6= 0, a001 +c011 = 0, then a1= 0, b16= 0 and a2 1+b2 16= 0. From Theorem 4.10, if system (5.17) admits some first integral then there exists an inverse Jacobi multiplier of the form M=h+···. From Lemma 4.11, αimust vanish for all 2i > 2. In this case, we have obtained α4=α6= 0 and α8=a5 001c101/32, whose vanishing implies that c101 = 0. With this hypothesis, we have obtained α2j= 0 for j= 2,...,20. This situation corresponds to item (ii) of the statement. (b2) If a001 6= 0,a001 + 2c011 = 0, then a16= 0, b1= 0 and a2 1+b2 16= 0. From Theorem 4.10, if system (5.17) admits some first integral then there exists an inverse Jacobi multiplier of the form M=h+···. Again, from Lemma 4.11 we obtain that α2jmust vanish for all j≥2. We have obtained α2j= 0 for j= 2,...,18 and we conjecture that system admits some analytic first integral. This situation corresponds to item (iii) of the statement. (b3) If a001(a001+c011)(a001+2c011)6= 0, then a1b16= 0 and using Theorem 4.10, we obtain that system (5.17) is integrable if, and only if, it admits some inverse integrating factor of the form M=h2+··· and there are n1, n2∈Ncoprime such that 2n1a1+2n2b1= 0, i.e., (n2−n1)a001 + (2n2−n1)c011 = 0. We notice that a001 =qc011 6= 0 where q=−2n2−n1 n2−n1∈Q. We claim that q < −2 or q > −1. Namely, if n2> n1then q=−2−n1 n2−n1<−2, whereas if n2< n1then q=−1 + n2 n1−n2>−1. Using this condition, from Lemma 4.11 we obtain that if system (5.17) is integrable then α2imust vanish for all i > 2. We have obtained α12 =−c7 011 c101 q4(q+ 2)(5q+ 8)/1920. We note that c011 6= 0, q6= 0, q+ 2 6= 0, 5q+ 8 6= 0 (otherwise, n1,n2are not coprime). Then, the vanishing of α12 occurs if, and only if, c101 = 0. Under this hypothesis, we have obtained α6=α8=α10 =α12 = 0 and α14 =c10 011 q4(3q+ 5) (q+ 2) (q+ 1) (84q3+ 201q2+ 129q+ 10)/49152, α16 = 0, α18 =c14 011 q5(q+ 2) (q+ 1) (308448q7+ 2217516q6+ 6661397q5+ 10859256q4 +10383887q3+ 5761860q2+ 1604516q+ 111300)/113246208. Using that α14 and α18 can not vanish simultaneously, we deduce that, in this situation, system (5.17) does not admit any first integral. Acknowledgements. 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