# On the Formal Integrability Problem for Planar Differential Systems

Authors: Antonio Algaba; Cristóbal García; Jaume Giné
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study the analytic integrability problem through the formal integrability problem and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. From the results obtained, we consider some families of analytic differential systems inℂ2, and imposing the formal integrability we find resonant centers obviating the computation of some necessary conditions.


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## Body

## 1. Introduction

One of the main open problems in the qualitative theory of differential systems inℝ2 is the distinction between a center and a focus, called the center problem, and its relation with the integrability problem; see, for instance, [1–5]. The notion of center can be extended to the case of a p:-q resonant singular point of a polynomial vector field in ℂ2 and to some other situations (resonant node, saddle node, and nonelementary singular points); see [6]. Recently, several works about this subject have appeared where the classification of the resonant centers for certain families is given using powerful computational facilities; see, for instance, [6–17].There exist several methods to find necessary conditions; see [1, 18]. However, there is no general method to provide the sufficiency for each family that satisfies some necessary conditions. The sufficiency is obtained verifying that the system is Hamiltonian, or that it has certain reversibility, or certain Lie symmetry, or finding a first integral well defined in a neighborhood of the singular point, sometimes finding an integrating factor that allows to construct a first integral, reducing the system to an integrable system; see, for instance, [4, 5, 18–23] and the references therein. These methods have proved ineffective in certain families that already verify some necessary conditions, and in many papers some cases are established as open problems; see, for instance, [7, 10, 11]. Carrying, in practice, to prove the sufficiency by ad hoc methods for each particular case where it has been possible, see [6–8, 12, 16, 17, 24–26]. One of the ways to prove the sufficiency is proving the existence of a formal first integral. The existence of a formal first integral implies the existence of an analytic first integral for an isolated singularity from the results obtained by Mattei and Moussu; see [27] and Theorem 2 in Section 2.Therefore, one of the main objectives for the next years will be to find an algorithm (if exists) or methods that give directly sufficient conditions to decide whether or not a differential system in the plane admits a formal first integral at a singular point.This paper is the first step in this direction. In this paper, we study the analytic integrability through the formal integrability and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. Moreover, from the results obtained in this work, we consider some families of analytic differential systems inℂ2, and imposing the formal integrability we find centers or resonant centers obviating the computation of the necessary conditions (see, for instance, Proposition 11).


## 2. Definitions and Preliminary Concepts

We first consider a real system of ordinary differential equations onℝ2 with an isolated singular point at the origin, whose linear parts are nonzero pure imaginary numbers. By a linear change of coordinates and a constant time rescaling, the system takes the form
(1)u˙=v+⋯,v˙=-u+⋯.
The classical Poincaré-Liapunov center theorem states that the origin is a center if and only if the system admits an analytic first integral of the form ϕ(u,v)=u2+v2+⋯; see, for instance, [18, 28–31] and the references therein.If system (1) is complexified in a natural way taking z=u+iv, then we obtain a differential equation of the form z˙=iz+⋯. In this case one constructs, step by step, the formal first integral Φ=zz-+⋯ satisfying the equation Φ˙=v3|z|4+v5|z|6+⋯, where the coefficients v2i+1, called the focus quantities, are polynomials in the coefficients of the original system. The theorem of Poincaré-Liapunov [32, 33] says that the point z=0 is a center if and only if all the v2i+1=0. Existence of a first integral ϕ is equivalent to existence of an analytic first integral for the complexified equation of the form Φ=zz-+⋯. Taking the complex conjugated equation, there arises an analytic system of ordinary differential equations on ℂ2 of the form z˙=iz+⋯, w˙=-iw+⋯ where w=u-iv. Hence, after the complexification, the system is transformed into an analytic system with eigenvalues +i and -i. This is the 1:-1 resonant singular point and the numbers v2i+1 become the coefficients of the resonant terms in its orbital normal form. Dulac [34] chosen this way to approach the center problem for quadratic systems.The next natural generalization is to consider the case of an analytic vector field inℂ2 with p:-q resonant elementary singular point
(2)x˙=px+⋯,y˙=-qy+⋯,
where p,q∈ℤ with p,q>0. These facts motivate the generalization of the concept of real center to certain classes of systems of ordinary differential equations on ℂ2. In this case, we have the following definition of a resonant center or focus, coming from Dulac [34]; see also [6].Definition 1.
Ap:-q resonant elementary singular point of an analytic system is a resonant center if, and only if, there exists a local meromorphic first integral Φ=h0+⋯, with h0=xqyp. This singular point is a resonant focus of order k if, and only if, there is a formal power series Φ=xqyp+⋯ with the property Φ˙=gkh0k+1+⋯.Recently, several works are focused on the study of resonant centers for complex analytic systems. In these works, the integrability and linearizability problems are studied. The linearizability problem is focused on the study of the existence of an analytic change of coordinates that linearize the complex analytic system.Mattei and Moussu [27] proved the next result for all isolated singularities.Theorem 2.
Assume that system (1) or (2) with an isolated singularity at the origin has a formal first integral H∈ℝ[[x,y]] around it. Then, there exists an analytic first integral around the singularity.In the light of the former result, we can conclude that in order to prove the existence of an analytic first integral we only need to prove the existence of a formal first integral. Therefore, the formal integrability problem takes a primary role for the upcoming investigations on the center and resonant center problem.


## 3. Blowup of Resonant Saddles

In this section, we will consider the first method to approach the formal integrability problem for resonant centers. We consider the resonant analytic system (2). We now do the blowup of this singularity. This means that we apply the map (x,y)→(x,z)=(x,y/x). The point x=y=0 is replaced by the line x=0, which contains two singular points that correspond to the separatrices of (2) and are the saddles: p1 which is (p+q):-p resonant and p2 which is (p+q):-q resonant. The following lemma is proved in [12] using the normal orbital form of a resonant analytic system (2).Lemma 3.
If one of the pointsp1 or p2 is orbitally analytically linearizable then the point x=y=0 is a p:-q resonant center.As a corollary of this lemma, we get the following useful property which can be used to approach the resonant center problem. If the pointp1 is orbitally analytically linearizable then it has a formal first integral in the variables (x,z) of the form
(3)H=xpzq+p∑n=0∞‍fn(x)zn,
with fn(0)≠0. Hence, the original p:-q resonant point has a nonformal first integral of the form
(4)x-qyq+p∑n=0∞‍fn(x)x-nyn=∑n=0∞‍fn(x)x-q-nyq+p+n.
Consequently, if the resonant point p:-q has a non formal first integral of the form (4) then by Lemma 3 the resonant point p:-q is a resonant center. The same result can be established for the other saddle p2. These types of results were also given by Bruno in [35, 36].


## 4. Theε-Method for Resonant Centers

We recall in this section theε-method developed in [23] which we apply here to resonant centers. We consider system (2) which we write into the form
(5)x˙=P(x,y)=px+F1(x,y),y˙=Q(x,y)=-qy+F2(x,y),
where F1(x,y) and F2(x,y) are analytic functions without constant and linear terms, defined in a neighborhood of the origin.To implement the algorithm we introduce a rescaling of the variables and a time rescaling given by(x,y,t)→(εp-x,εq-y,εr-t) where ε>0 and p-, q-, and r-∈ℤ and system (5) takes the form
(6)x˙=εr--p-(pεp-x+F1(εp-x,εq-y)),y˙=εr--q-(-qεq-y+F2(εp-x,εq-y)).
We choose p-, q-, r- in such a way that system (6) will be analytic in ε. Hence, by the classical theorem of the analytic dependence with respect to the parameters, we have that system (6) admits a first integral which can be developed in power of series of ε because it is analytic with respect to this parameter. Therefore, we can propose the following development for the first integral:
(7)H(x,y)=∑k=0∞‍εkhk(x,y),
where hk(x,y) are arbitrary functions. We notice that F1(x,y) and F2(x,y) are analytic functions, where both can be null and F1(0,0)=F2(0,0)=0, so we can develop them, in a neighborhood of the origin, as convergent series of x and y of the form
(8)F1(x,y)=pn(x,y)+pn+1(x,y)+⋯+pj(x,y)+⋯,F2(x,y)=qn(x,y)+qn+1(x,y)+⋯+qj(x,y)+⋯,
with n=min{subdeg|(0,0)F1(x,y),subdeg|(0,0)F2(x,y)}≥2. We recall that given an analytic function f(x,y) defined in a neighborhood of a point (x0,y0), we define subdeg|(x0,y0)f(x,y) as the least positive integer j such that some derivative (∂jf/∂xi∂yj-i)(x0,y0) is not zero. We notice that this computation depends on the variables (x,y) on which the function f(x,y) depends, so we will explicit the variables used in each computation of subdeg. For instance, subdeg|(x0,y0)f(x,y)=0, if and only if f(x0,y0)≠0. In (8), pj(x,y) and qj(x,y) denote homogeneous polynomials of x and y of degree j≥n. It is possible that pn(x,y) or qn(x,y) be null but, by definition, not both of them can be null. The simplest case is to consider in the rescaling p-=q-=1. In fact, this case is equivalent to impose that system (8) has a first integral which can be expanded as a formal series in homogeneous parts.The richness of theε-method is that, using the parameter ε, the functions hk(x,y) need not be homogeneous parts and we can construct also a singular series expansion in the variables x and y; see [23]. The method depends heavily on h0 the first integral of the initial quasi-homogeneous system. The simpler is h0 farther we go with the method. However, h0 can be chosen using different scalings of variables (x,y,t)→(εp-x,εq-y,εr-t) where ε>0 and p-, q-, and r-∈ℤ, in such a way that h0 will be as simple as possible. The method gives necessary conditions to have analytic integrability or a singular series expansion around a singular point and information about what is called in [23] the essential variables of a system. In the method developed in [23], the parameter ε needs not be small. The parameter ε may be relatively large (for instance ε→1). The convergence of series (7) with respect to (x,y) must be analyzed in each particular case, and the convergent rate depends upon the nonlinear terms of the system (8). In the case of a resonant center, the most convenient h0 is xqyp.From now on we suppose that system (5) admits a formal first integral. Therefore, hk(x,y) are homogeneous polynomials of degree k and we can take h0(x,y)=0. In this case if we impose that series (7) be a formal first integral of system (5) we obtain at each order of homogeneity
(9)px∂hk∂x(x,y)-qy∂hk∂y(x,y)+Gk(x,y)=0,
where Gk(x,y) depends on the previous hi for i=1,…,k-1. The solution of the partial equation (9) is
(10)hk(x,y)=F(xq/py)-∫xGk(s,(xq/py)/sq/p)psds,
where F is an arbitrary function of xq/py. We require that hk be a polynomial and therefore an hk not having logarithmic terms. In order to avoid the logarithmic terms, it is easy to see that the function Gk(x,y) must not have polynomial terms of the form (xqyp)m for m∈ℕ. This result is the same as that we obtain using normal form theory; see [6].We must impose the vanishing of all the logarithmic terms (if there exists any difference from zero) and prove that all the functionshi(x,y) for i>1 are polynomials. In fact we must prove this by induction assuming that all the hi for i=1,…k-1 are polynomials, and prove that the recursive partial differential equation with respect to hk (see (9)) gives also a polynomial. This is not so easy and even less working with a partial recursive differential equation as (9).At this point we think in the blow-upz=y/x because each hi(x,y) are homogeneous polynomials of degree i. This blowup transforms the system (5) into a system of variables (x,z) of the form
(11)z˙=-(p+q)z+xℱ(x,z),x˙=px+x2𝒢(x,z),
where ℱ(0,0)=0. After that we propose a formal first integral of the form
(12)H~=∑i≥1∞‍fi(z)xi,
where fi(z) must be polynomials of degree ≤i (if the logarithmic terms are zero). If we impose that series (12) be a first integral of the system in variables (x,z), we obtain that the term b20x2 must be zero. However, we can vanish this term doing the change y-=y-αx2 and x-=x, with α=-b20/(2p+q). We must also take f1(z)=0. Finally at each power of y we have the recursive equation
(13)kpfk(z)-(p+q)zfk′(z)+gk(z)=0,
where gk(z) depends of the previous functions fi for i=1,…,k-1. Hence in this case instead of getting the partial recursive differential equation (9), we obtain an ordinary recursive differential equation whose solution is given by
(14)fk(z)=C~kzkp/(p+q)+zkp/(p+q)∫zs-1-kp/(p+q)p+qgk(s)ds,
where C~k is an arbitrary constant. However, we are going to see that, although we place in a resonant center where all the fk be polynomials, this blowup does not allow us to prove, in general, by induction of the existence of a formal first integral of the form (12).In this sense, consider the recursive differential equation (13) whose solution is (14) and we assume that system (5) has a resonant degenerate center at the origin. It is easy to see that there always exists a value k0 such that for k≥k0 the arbitrary polynomials fi(z) for i=1,…,k0-1 can gives following the recursive equation (13) logarithmic terms. Therefore, we can not apply the induction method to prove that the recursive equation (13) gives always a polynomial.To see this we must to see that the solution (14) of the recursive equation (13) can give a logarithmic term. This happens when
(15)-1-k0pp+q+mk=-1,
where mk is the degree of the polynomial gk(s). From here we have that k0=mk(p+q)/p. Hence if p=1 then k0=mk(1+q) which can be satisfied because mk and q are positive integers. For the case p≠1 and taking into account that the value of mk increases when k increases, it can also exist a value of mk such that mk is divisible by p and that gives the value of k0 that can give logarithmic terms.Therefore the conclusion is that the formal construction of the first integral (7) using homogeneous terms or using the blow-up z=y/x and the formal series (12) do not allow to use the induction method in order to verify the existence of a formal first integral. We must use other developments which is the subject of the next section.


## 5. Other Developments of the Formal First Integral

In this section, we consider other developments of the formal first integral. We consider the formal development of the first integral of system (5) in a series in the variable x or in y; that is, we consider
(16)H1=∑k=0∞‍fk(x)yk,orH2=∑k=0∞‍gk(y)xk.First we consider a general analytic system that we can always write into the following forms:(17)x˙=f(x)+yΦ1(x,y),y˙=g(x)+yΦ2(x,y),
or
(18)x˙=f(y)+xΨ1(x,y),y˙=g(y)+xΨ2(x,y),
where f, g, Φ1, Φ2, Ψ1, and Ψ2 are analytic in their variables. For systems (17) and (18), we have the following straightforward result.Proposition 4.
If we impose that the seriesH1 (H2, resp.) be a first integral of system (17) (system (18), resp.) among others, the first condition is f0′(x)f(x)+f1(x)g(x)=0, (f0′(y)g(y)+f1(y)f(y)=0, resp.).In order that this condition generates a collection of recursive differential equations where eachfi(x) does not depend on fi+1(x), we must impose g(x)=0 (f(y)=0, resp.) which implies that y=0 (x=0 resp.) is an invariant algebraic curve of system (17) (system (18) resp.).In fact for system (5), there are always a new coordinates (z1,z2) where z1=0 and z2=0 are invariant curves. These invariant curves are defined by the stable and instable manifold of the p:-q resonant singular point and therefore the previous conditions are directly satisfied. In these coordinates any p:-q resonant singular point is a Lotka-Volterra system. In the following result, we are going to see that we always can find a new coordinate where y=0 is an invariant algebraic curve of the transformed system (5).We consider system (2) which we write into the form
(19)x˙=P(x,y)=px(1+P0(x))+∑j≥1‍Pj(x)yj,y˙=Q(x,y)=xQ0(x)-qy(1+Q1(x))+∑j≥2‍Qj(x)yj,
where Pj(x)=∑i≥0‍aijxi and Qj(x)=∑i≥0‍bijxi are analytic functions, defined in a neighborhood of the origin.Lemma 5.
System (19) is orbitally equivalent to
(20)x˙=P(x,y)=px(1+P0(x))+∑j≥1‍Pj(x)yj,y˙=Q(x,y)=-qy+∑j≥2‍Qj(x)yj,
that is, Q0(x)≡Q1(x)≡0. Moreover, (p,q) are coprimes.Proof.
First we prove that we can achieveQ0(x)≡0. Assume that Q0(x)≢0 and let N=min{i∈ℕ∪{0}:bi0≠0}. The change of variable u=x, v=y-(bN0/(p(N+1)+q))xN+1 transforms system (19) into
(21)u˙=pu(1+P~0(u))+∑j≥1‍P~j(u)vj,v˙=uQ~0(u)-qv(1+Q~1(u))+∑j≥2‍Q~j(u)vj,
where Q~0(u)=∑i≥0‍b~i0ui and b~i0=0 for i=0,1,…,N because v˙=y˙-((N+1)bN0/(p(N+1)+q))uNx˙ hence
(22)uQ~0(u)=bN0uN+1-qbN0p(N+1)+quN+1-p(N+1)bN0p(N+1)+quN+1+𝒪(uN+2)=𝒪(uN+2).
Therefore, by means of successive change of variables we can transform system (19) into
(23)u˙=pu(1+P^0(u))+∑j≥1‍P^j(u)yj,v˙=-qv(1+Q^1(u))+∑j≥2‍Q^j(u)vj.
To complete the proof it is enough to apply the scaling dt=dT/(1+Q^1(u)).Lemma 6.
Let𝒢 be a formal function with 𝒢(0)=0. The system
(24)(x˙y˙)=(px-qy)+(pxqy)𝒢(xqyp),
is formally integrable if and only if 𝒢≡0.Proof.
The sufficient condition is trivial sincexqyp is a first integral of (x˙,y˙)T=(px,-qy)T.
We now will prove the necessary condition. LetF(x,y)=(px,-qy)T+(px,qy)T𝒢(h), where h=xqyp and 𝒢(h)=∑j≥1‍gjhj. 𝒢≢0; otherwise, the proof is finished. We consider j0=min{j∈N∣gj≠0}. If I is a first integral of this system, then there exists M∈ℕ such that I=hM+⋯ and 0=∇I·F=2pqgj0hj0+1+⋯. Therefore, gj0=0 and this is a contradiction.The next result shows that any integrable system (20) admits always a first integral of the form H1.Proposition 7.
If system (20) is formally integrable then H(x,y)=∑j≥0‍fj(x)yj with fj(x) a formal function is a first integral and moreover f0=⋯fp-1=0.Proof.
By applying the Poincare-Dulac normal form, there exists a change of variable to transform (20) into
(25)(u˙v˙)=(pu-qv)+(∑k≥1akuhk∑k≥1bkvhk),
where h=uqvp. We can assume that the change of variable is u=x+⋯, v=y(1+⋯) because the axis y=0 and v=0 are invariant.
Considering the formal functionsℱ(h)=∑k≥1‍ckhk and 𝒢~(h)=∑k≥1‍dkhk with ck=(qak-pbk)/2pq and dk=(qak+pbk)/2pq, we get
(26)(u˙v˙)=(pu-qv)(1+ℱ(uqvp))+(puqv)𝒢~(uqvp),
where ℱ, 𝒢~ are formal functions with ℱ(0)=𝒢~(0)=0.
Moreover, by the scaling of the time,dt=(1+ℱ)dτ, we can get ℱ≡0; that is, it is possible to transform (20) into
(27)(u′v′)=(pu-qv)+(puqv)𝒢(uqvp),
where 𝒢=𝒢~/(1+ℱ) is a formal function with 𝒢(0)=0.
If this system is formally integrable, applying Lemma6, we obtain 𝒢=0. Therefore, H(x,y)=(x+⋯)qyp(1+⋯)p=∑k≥p‍fk(x)yk is a first integral of system (20).The main result of this work is as follows.Theorem 8.
System (20) admits an analytic first integral H(x,y)=∑k≥p‍fk(x)yk if for each k∈ℕ such that k=Mp with M∈ℕ is verified that the derivative g^k(Mq)(0)=0, where g^k for k≥p is the analytical function(28)g^k(x):=gk(x)p(1+P0(x))f-p(x)k,f-p(x)=exp(-∫0x‍(qP0(s)ps(1+P0(s)))ds),andfork>k>kkkgp=0,andfork>pgk(x)=∑j=1k-p‍((k-j)fk-j(x)Qj+1(x)+fk-j′(x)Pj(x)),
and for k≥p(29)fk(x)=xkq/pf-p(x)k(Ck-∫xs-1-(kq/p)g^k(s)ds).Proof.
By Proposition7, we have f0=⋯=fp-1=0. If we impose that H(x,y) be a first integral of system (20), we obtain that the first condition is
(30)-pqfp(x)+pxfp′(x)(1+P0(x))=0.
The next coefficients for each power of y are of the form
(31)-qkfk(x)+px(1+P0(x))fk′(x)+gk(x)=0,
where gp=0 and gk(x) for k>p depends of the previous fi(x) for i=p, p+1,…,k-1. More specifically we obtain the expression (28).
The solution of homogeneous equation associated to (31) is
(32)fkh(x)=exp(∫‍kqpx(1+P0(x))).
We can define the following analytical function b(x)=-(q/px)(P0(x)/(1+P0(x))) and the integrand of the previous expression of fkh is a rational function which admits the following fraction decomposition:
(33)kqpx(1+P0(x))=kqpx+kb(x).
Therefore, fkh(x)=xkq/pexp(∫‍kb(x))=xkq/p(exp(∫0x‍b(s)ds))k and for the first equation which corresponds to k=p we have that gp(x)=0. Therefore, fph(x) has the form fph(x)=Cpxq(f-p(x))p where f-p(x)=exp(∫0x‍b(s)ds) is an analytic function with f-p(0)=1. So, fkh(x)=Ckxkq/pf-p(x)k and the solution of (31) is given by
(34)fk(x)=xkq/pf-p(x)k(Ck-∫0x‍s-kq/pf-p(s)-kps(1+P0(s))gk(s)ds).
Taking into account the form of gk(x), we obtain the expression (29) for fk. Now it is straightforward to see that the integral does not give logarithmic terms in the case kq/p∉ℕ and in the case k=Mp because g^k(Mq)(0)=0. If we choose Ck=0, we have that each fk(x) for all k is an analytic function and therefore system (20) under the assumptions of the theorem admits a formal first integral of the form H(x,y)=∑k≥p‍fk(x)yk.Corollary 9.
System (20) admits a formal first integral of the form H(x,y)=∑j≥p‍fj(x)yj if one of the following conditions holds: (a)
P0(x)≡0 and for each k=Mp, M∈ℕ, gk(x) is a polynomial of degree at most Mq-1;(b)
P0(x)≡axr, with r a positive integer and for each k=Mp, M∈ℕ, gk(x) is a rational function of the form gk(x)=g~k(x)/(1+axr)-1+q/rp where g~k is a polynomial of degree at most Mq-1.Proof.
In case (a) it is enough to apply Theorem8 for f-p(x)≡1 and g^k(x)=gk(x)/p. For k=Mp, M∈ℕ and we have g^k(Mq)(x)=gk(Mq)/p=0, because the degree of gk(x) is less than qM.
In case (b), we have thatf-p(x)≡(1+axr)-q/rp, and g^k(x)=gk(x)/p(1+axr)1-q/rp. Therefore, g^k(x)=g~k(x)/p and g^k(Mp)(x)=0 because the degree of g~k(x) is less than qM. We obtain the result by applying also Theorem 8.Notice that in fact statement (a) is contained in statement (b). However, we give the two statements in a separate form in order to be applied directly to different examples.Lemma 10.
If system (20) with P0(x)≡0 admits a formal first integral of the form H(x,y)=∑j≥p‍fj(x)yj then the following conditions hold. (a)
If for eachk such that kq/p∉ℕ, gk is a polynomial then it is possible to choose fk polynomial with deg(fk)=deg(gk).(b)
If for eachk=Mp, M∈ℕ, gk(x) is a polynomial with deg(gk)≤Mq-1 then it is possible to choose fk polynomial with deg(fk)≤Mq-1.Proof.
By applying Theorem8, we obtain f-p(x)≡1 and g^k(x)=gk(x)/p. If we choose Ck=0 in (29), we obtain
(35)fk(x)=-1pxkq/p∫xs-1-kq/pgk(s)ds.
In case (a), the result is trivial since the integral does not give logarithmic terms because kq/p∉ℕ.
In case (b),k=Mp and fk(x)=-(1/p)xMq ∫xs-1-Mqgk(s)ds. Therefore, we obtain the result applying that gk is a polynomial with deg(gk)≤Mq-1.Based on the results presented in this work, the following proposition gives a large family of analytic systems that have an analytic first integral.Proposition 11.
The analytic system(36)x˙=px+∑j≥1‍Pj(x)yj,x˙=-qy+∑j≥2‍Qj(x)yj,
where Pj and Qj are polynomials such that deg(Pj)≤⌊jq/p⌋, deg(Qj)≤⌊(j-1)q/p⌋-1 has an analytic first integral in a neighborhood of the origin where ⌊a⌋ is the integer part of a∈ℝ.Proof.
From Proposition7, if system (36) is integrable then a first integral is of the form H(x,y)=∑j≥p‍fj(x)yj where fj(x) is a formal function. Moreover, P0(x)≡0 and, therefore, by applying Theorem 8 we obtain f-p(x)≡1, g^k(x)=gk(x)/p, gp=0 and fp(x)=xq.
We will prove now by induction thatdeg(fj)≤⌊jq/p⌋ and deg(gj)≤⌊jq/p⌋-1 for all j∈ℕ and p≤j.
Forj=p is true since fp(x)=xq and gp=0. If we suppose that the hypothesis is true for 1≤j≤j0-1, we have to prove that deg(gj0)≤⌊j0q/p⌋-1 and deg(fj0)≤⌊j0q/p⌋.
On the other hand, applying (28), we have
(37)deg(gj0)≤max1≤j≤j0-p{deg(fj0-j)+deg(Qj+1),deg(fj0-j′)+deg(Pj)}≤⌊j0qp⌋-1,
since for 1≤j≤j0-p, we have
(38)deg(fj0-j)+deg(Qj+1)≤⌊(j0-j)q/p⌋+⌊jqp⌋-1≤⌊j0qp⌋-1,deg(fj0-j′)+deg(Pj)≤⌊(j0-j)qp⌋-1+⌊jqp⌋≤⌊j0qp⌋-1.
By applying Lemma10, we have that deg(fj0)≤⌊j0q/p⌋-1. In particular, deg(fj0)≤⌊j0q/p⌋.
Therefore, ifk=Mp, deg(gk)≤Mq-1, gkMq(0)=0 and g^kMq(0)=0.
To finish the proof, it is enough to apply Theorem8.However, as it is difficult to determine the new coordinates where system (19) has Q0(x)≡Q1(x)≡0, we are going to work with the original system (5), imposing directly these two conditions. In order that one of the series H1 or H2 be a formal first integral of system (5), we have that these series must have the term xqyp as a first monomial in its development. We will see that in this case and under some conditions we can use the induction method in order to verify the existence of a formal first integral.The following result is only established for the seriesH1 but we can obtain a similar result for the case where H2 is a formal first integral of system (5).Proposition 12.
We consider system (5) where we write F1 and F2 into the form
(39)F1(x,y)=∑i+j=2∞‍aijxiyj,F2(x,y)=∑i+j=2∞‍bijxiyj,
where aij and bij are arbitrary constants. If we impose that H1 be a first integral of system (5), we obtain that the first condition is
(40)∑i=2∞‍bi0xif1(x)+pxf0′(x)+∑i=2∞‍ai0xif0′(x)=0.The proof of this proposition is also straightforward developing in powers ofy and taking the coefficient of the power y0. Hence in order to have a recursive differential equations where each fi(x) does not depend on fi+1(x), we must impose that bi0=0 for all i and consequently f0 must be a constant that we can take without loss of generality equal to zero. The consequence, as before, is that if all bi0=0 for all i the system (5) has y as invariant algebraic curve.The next coefficients for each power ofy are of the form
(41)k(-q+∑i=1∞‍bi1xi)fk(x)+(px+∑i=2∞‍ai0xi)fk′(x)+gk(x)=0,
where gk(x) depends of the previous fi(x) for i=1,2,…,k-1, and for the first equation which corresponds to k=1 we have that g1(x)=0. Therefore, f1(x) has the form
(42)f1(x)=exp(-∫‍-q+∑i=1∞‍bi1xipx+∑i=2∞‍ai0xi).
The integrand of the previous expression of f1 is a rational function which admits the following fraction decomposition:
(43)-q+∑i=1∞‍bi1xipx+∑i=2∞‍ai0xi=Ax+B(x)p+∑i=2∞‍ai0xi-1,
where A=-q/p and B(x) is a formal series. Therefore, f1(x)=xq/pf-1(x) where f-1 is the corresponding formal series obtained after the integration. Now we consider the particular case p=1. The following theorem gives sufficient conditions to have formal integrability in this case. A similar proposition can be established for the case when the first integral is of the form H2.Proposition 13.
System (5) where we write F1 and F2 into the form (39) admits a formal first integral of the form H1 if p=1, bi0=0 for i≥2 and bi1=0 for i≥1 and
(44)gk(x)=g~k(x)P(x,0)xf-1(x)-kwithg~k(kq)(0)=0.Proof.
Under the assumptions of the theorem, we have thatf0=0, f1=xqf-1(x), and (41) takes the form
(45)-kqfk(x)+P(x,0)fk′(x)+gk(x)=0,
whose solution is given by
(46)fk(x)=xkqf-1(x)k(Ck-∫xs-kqf-1(s)-kP(s,0)gk(s)ds).
Taking into account the form of gk(x), we obtain
(47)fk(x)=xkqf-1(x)k(Ck-∫xs-1-kqg~k(s)ds).
Now it is straightforward to see that the integral does not give logarithmic terms because the kq-th derivative of g~k(s) at zero is zero; that is, g~k(kq)(0)=0. We have that each fk(x) for all k is an analytic function and therefore system (5) under the assumptions of the theorem admits a formal first integral of the form (39).Examples where Proposition13 can be applied can be found in [7, 12, 16, 17] where some partial results are given. We also can establish the following corollary in the case where the gk(x) are polynomials or some irrational functions.Corollary 14.
System (5) where we write F1 and F2 into the form (39) admits a formal first integral of the form H1 if one of the following conditions holds: (a)
p=1, ai0=bi0=0 for i≥2 and bi1=0 for i≥1 and gk(x) is a polynomial of degree at most kq-1;(b)
p=1, ai0=0 for i≠r+1, bi0=0 for i≥2 and bi1=0 for i≥1 with r a positive integer and gk(x) is a rational function of the form gk(x)=g~k(x)/(1+axr)-1+kq/r where g~k is a polynomial of degree at most kq-1.Proof.
In case (a) for we havef0=0 and f1=xq (41) take the form
(48)-kqfk(x)+xfk′(x)+gk(x)=0,
whose solution is given by
(49)fk(x)=Ckxkq-xkq∫xs-1-kqgk(s)ds,
and taking into account that gk(x), that depends on fi for i=1,2,…,k-1, is at most of degree kq-1, we obtain that fk is a polynomial. Moreover, in this case system (5) becomes
(50)x˙=px+yϕ1(x,y),y˙=-qy+yϕ2(x,y),
where ϕ1 and ϕ2 are analytic functions.
In case (b), we have thatf0=0, f1=xq(1+axr)-q/r and (41) takes the form
(51)-kqfk(x)+xfk′(x)+axr+1fk′(x)+gk(x)=0,
whose solution is given by
(52)fk(x)=xkq(1+axr)-kq/r×(Ck-∫xs-1-kq(1+as)-1+kq/rgk(s)ds),
and taking into account the form of gk(x), we obtain that fk(x) is also of the form fk(x)=pk(x)/(1+axr)-kq/s where pk(x) is a polynomial because the integral in (52) does not give logarithmic terms.Notice that in fact statement (a) is contained in statement (b). However, we give the two statements in a separate form in order to be applied directly to different examples. Corollary14 generalizes a lot of cases obtained in the literature when some concrete families of polynomials systems were studied. We have not made an exhaustive study of all the cases included in these large families but for instance contains the cases (3) and (4) of Theorem  3 in [17] and case (3) of Theorem  2 in [16].In resume, when we study a family that satisfies the necessary conditions, we must impose if the family system has a formal first integral of the formH1 or H2. If none of them work, we can look for the proper coordinates (z1,z2) defined by the stable and instable manifold of the p:-q resonant singular point. In this way, we must study if at least one of the separatrices of the resonant point is algebraic. The best case is when both are algebraic. Therefore, the problem to find proper coordinates (z1,z2) becomes a problem of finding the invariant algebraic curves of system (5) passing through the p:-q resonant singular point. The existence of invariant algebraic curves takes a leading role in the formal integrability theory and is the base to find the proper coordinates system. This is shown in the examples provided in [12] that are solved by ad hoc methods and where the changes of variables proposed to solve the examples are always invariant algebraic curves of the original system passing through the origin. In fact, in [12], it recalled the Abhyankar-Moch theorem [37]. This theorem establishes, assuming the existence of an invariant rational algebraic curve passing through the origin, the existence of a rational invertible change of variables such that the invariant curve becomes one of the invariant axes. In this paper, this result is generalized in the sense that any p:-q resonant singular point is formal orbitally equivalent to a system with y=0 as invariant algebraic curve; see Lemma 5.


## 6. Examples

In this section, we present some examples where the results developed in this paper are applied.Example 15.
We first consider the1:-1 resonant quadratic system given by
(53)x˙=x+a20x2+a11xy+a02y2,y˙=-y+b20x2+b11xy+b02y2.
The aim is not to do an exhaustive study of when system (53) has a formal first integral in a neighborhood of the origin. In fact, the complex center cases were studied by Dulac; see [34]. The aim is only to show how to use the results given in this paper. If we apply directly Corollary 14 statement (a), we obtain b20=0 (this condition implies that y=0 be an algebraic invariant curve of system (53)) and a20=b11=0. In this case, the quadratic system (53) reads
(54)x˙=x+a11xy+a02y2,y˙=-y+b02y2
and admits a formal first integral, and by Theorem 2 system (54) has an analytic first integral in a neighborhood of the origin.
In fact, the associated equation to system (54) is a linear equation with respect to x and the system has a Darboux first integral of the form
(55)H(x,y)=(b02y-1)-1-(a11/b02)×[2a02-2(a02a11+a02b02)y×+(a113-a11b022)xy+(a02a112+a02a11b02)y2].Example 16.
Consider the 1 : −1 resonant Lotka–Volterra planar complex quartic system given by(56)x˙=x(1-a10x-a01y-a20x2-a11xy-a02y2-a30x3-a21x2y-a12xy2-a03y3),y˙=-y(1-b10x-b01y-b20x2-b11xy-b02y2-b30x3-b21x2y-b12xy2-b03y3).
The Lotka–Volterra planar complex systems have been studied in several papers; see [7, 11, 13–15]. In any case, the Lotka–Volterra planar complex quartic system is an open problem computationally infeasible. However, with the results of this paper we are going to see that we can give some sufficient conditions to have an analytic first integral without computing any resonant focus value. We propose a formal first integral of the form H1, that is, of the form (16). The recurrence that gives fi(x) is given by -ifi(x)+xfi′(x)+gi(x)=0 where gi(x) is(57)(i-3)b03fi-3(x)+(i-2)b02fi-2(x)+(i-2)b12xfi-2(x)+(i-1)b01fi-1(x)+(i-1)b11xfi-1(x)+(i-1)b21x2fi-1(x)+ib10xfi(x)+ib20x2fi(x)+ib30x3fi(x)-a03xfi-3′(x)-a02xfi-2′(x)-a12x2fi-2′(x)-a01xfi-1′(x)-a11x2fi-1′(x)-a10x2fi′(x)-a20x3fi′(x)-a30x4fi′(x).
Now we impose that gi(x) must be of degree ≤i-1. This implies that a10=a20=a30=b10=b20=b30=0 and b21=a21=b11=a11=0 and in this case the recurrence is -ifi(x)+xfi′(x)+gi(x)=0 where gi(x) is given by
(58)(i-3)b03fi-3(x)+(i-2)(b02+b12x)fi-2(x)+(i-1)b01fi-1(x)-a03xfi-3′(x)-a02xfi-2′(x)-a12x2fi-2′(x)-a01xfi-1′(x).
We obtain that its solution is given by
(59)fi(x)=Cixi-xi∫xs-1-igi(s)ds,
and taking into account that gi(x) is of degree ≤i-1, we can establish the following result.Proposition 17.
The system(60)x˙=x(1-a01y-a12xy2-a03y3),y˙=-y(1-b01y-b02y2-b12xy2-b03y3)
has an analytic first integral in a neighborhood of the origin.In fact, Proposition17 is a consequence of Proposition 11. This technique can be applied to several families of polynomial systems and in fact gives a new mechanism to obtain resonant centers and also real centers for the complex systems that have a real preimage (when this preimage exists, see e.g., [18]).



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