On the Formal Integrability Problem for Planar Differential Systems

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 in ℂ², and imposing the formal integrability we find resonant centers obviating the computation of some necessary conditions.

1. Introduction

One of the main open problems in the qualitative theory of differential systems in ℝ² 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 in ℂ², 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 on ℝ² 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 𝑢̇=𝑣+⋯,𝑣̇=−𝑢+⋯. (1) 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 𝑧̇=𝑖𝑧+⋯. In this case one constructs, step by step, the formal first integral Φ=𝑧𝑧+⋯ satisfying the equation Φ̇=𝑣₃|𝑧|⁴+𝑣₅|𝑧|⁶+⋯, where the coefficients 𝑣_2𝑖+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 𝑣_2𝑖+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 𝑧̇=𝑖𝑧+⋯, 𝑤̇=−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 𝑣_2𝑖+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 in ℂ² with 𝑝∶−𝑞 resonant elementary singular point 𝑥̇=𝑝𝑥+⋯,𝑦̇=−𝑞𝑦+⋯, (2) 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. A 𝑝∶−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=𝑥^𝑞𝑦^𝑝. This singular point is a resonant focus of order 𝑘 if, and only if, there is a formal power series Φ=𝑥^𝑞𝑦^𝑝+⋯ with the property Φ̇=𝑔_𝑘ℎ₀^𝑘+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 points 𝑝₁ 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 point 𝑝₁ is orbitally analytically linearizable then it has a formal first integral in the variables (𝑥,𝑧) of the form 𝐻=𝑥^𝑝𝑧^∞𝑞+𝑝_𝑛=0𝑓_𝑛(𝑥)𝑧^𝑛, (3) with 𝑓_𝑛(0)≠0. Hence, the original 𝑝∶−𝑞 resonant point has a nonformal first integral of the form 𝑥^−𝑞𝑦^∞𝑞+𝑝_𝑛=0𝑓_𝑛(𝑥)𝑥^−𝑛𝑦^𝑛=^∞_𝑛=0𝑓_𝑛(𝑥)𝑥^{−𝑞−𝑛}𝑦^{𝑞+𝑝+𝑛}. (4) 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. The 𝜀-Method for Resonant Centers

We recall in this section the 𝜀-method developed in [23] which we apply here to resonant centers. We consider system (2) which we write into the form 𝑥̇=𝑃𝑥,𝑦=𝑝𝑥+𝐹₁,̇𝑥,𝑦𝑦=𝑄𝑥,𝑦=−𝑞𝑦+𝐹₂𝑥,𝑦, (5) 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)→(𝜀^𝑝𝑥,𝜀^𝑞𝑦,𝜀^𝑟𝑡) where 𝜀>0 and 𝑝, 𝑞, and 𝑟∈ℤ and system (5) takes the form 𝑥̇=𝜀^𝑟−𝑝𝑝𝜀^𝑝𝑥+𝐹₁𝜀^𝑝𝑥,𝜀^𝑞𝑦,𝑦̇=𝜀^𝑟−𝑞−𝑞𝜀^𝑞𝑦+𝐹₂𝜀^𝑝𝑥,𝜀^𝑞𝑦. (6) 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: 𝐻𝑥,𝑦=^∞_𝑘=0𝜀^𝑘ℎ_𝑘𝑥,𝑦, (7) where ℎ_𝑘(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 𝐹₁𝑥,𝑦=𝑝_𝑛𝑥,𝑦+𝑝_𝑛+1𝑥,𝑦+⋯+𝑝_𝑗𝐹𝑥,𝑦+⋯,₂𝑥,𝑦=𝑞_𝑛𝑥,𝑦+𝑞_𝑛+1𝑥,𝑦+⋯+𝑞_𝑗𝑥,𝑦+⋯, (8) 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/𝜕𝑥^𝑖𝜕𝑦^𝑗−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), 𝑝_𝑗(𝑥,𝑦) and 𝑞_𝑗(𝑥,𝑦) denote homogeneous polynomials of 𝑥 and 𝑦 of degree 𝑗≥𝑛. It is possible that 𝑝_𝑛(𝑥,𝑦) or 𝑞_𝑛(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 the 𝜀-method is that, using the parameter 𝜀, the functions ℎ_𝑘(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)→(𝜀^𝑝𝑥,𝜀^𝑞𝑦,𝜀^𝑟𝑡) 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 𝑥^𝑞𝑦^𝑝.

From now on we suppose that system (5) admits a formal first integral. Therefore, ℎ_𝑘(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 𝑝𝑥𝜕ℎ_𝑘𝜕𝑥𝑥,𝑦−𝑞𝑦𝜕ℎ_𝑘𝜕𝑦𝑥,𝑦+𝐺_𝑘𝑥,𝑦=0, (9) where 𝐺_𝑘(𝑥,𝑦) depends on the previous ℎ_𝑖 for 𝑖=1,…,𝑘−1. The solution of the partial equation (9) is ℎ_𝑘𝑥,𝑦=𝐹𝑥^𝑞/𝑝𝑦−^𝑥𝐺_𝑘𝑠,𝑥^𝑞/𝑝𝑦/𝑠^𝑞/𝑝𝑝𝑠𝑑𝑠, (10) where 𝐹 is an arbitrary function of 𝑥^𝑞/𝑝𝑦. We require that ℎ_𝑘 be a polynomial and therefore an ℎ_𝑘 not having logarithmic terms. In order to avoid the logarithmic terms, it is easy to see that the function 𝐺_𝑘(𝑥,𝑦) must not have polynomial terms of the form (𝑥^𝑞𝑦^𝑝)^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 functions ℎ_𝑖(𝑥,𝑦) for 𝑖>1 are polynomials. In fact we must prove this by induction assuming that all the ℎ_𝑖 for 𝑖=1,…𝑘−1 are polynomials, and prove that the recursive partial differential equation with respect to ℎ_𝑘 (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-up 𝑧=𝑦/𝑥 because each ℎ_𝑖(𝑥,𝑦) are homogeneous polynomials of degree 𝑖. This blowup transforms the system (5) into a system of variables (𝑥,𝑧) of the form ̇(),̇𝑧=−𝑝+𝑞𝑧+𝑥ℱ𝑥,𝑧𝑥=𝑝𝑥+𝑥²𝒢(𝑥,𝑧), (11) where ℱ(0,0)=0. After that we propose a formal first integral of the form 𝐻^∼=^∞_𝑖≥1𝑓_𝑖(𝑧)𝑥^𝑖, (12) where 𝑓_𝑖(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 𝑘𝑝𝑓_𝑘(𝑧)−𝑝+𝑞𝑧𝑓^′_𝑘(𝑧)+𝑔_𝑘(𝑧)=0, (13) where 𝑔_𝑘(𝑧) depends of the previous functions 𝑓_𝑖 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 𝑓_𝑘(𝑧)=𝐶^∼_𝑘𝑧^{𝑘𝑝/(𝑝+𝑞)}+𝑧^{𝑘𝑝/(𝑝+𝑞)𝑧}𝑠^{−1−𝑘𝑝/(𝑝+𝑞)}𝑔𝑝+𝑞^𝑘(𝑠)𝑑𝑠, (14) where 𝐶^∼_𝑘 is an arbitrary constant. However, we are going to see that, although we place in a resonant center where all the 𝑓_𝑘 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 𝑓_𝑖(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 𝑘−1−⁰𝑝𝑝+𝑞+𝑚_𝑘=−1, (15) where 𝑚_𝑘 is the degree of the polynomial 𝑔_𝑘(𝑠). From here we have that 𝑘₀=𝑚_𝑘(𝑝+𝑞)/𝑝. Hence if 𝑝=1 then 𝑘₀=𝑚_𝑘(1+𝑞) which can be satisfied because 𝑚_𝑘 and 𝑞 are positive integers. For the case 𝑝≠1 and taking into account that the value of 𝑚_𝑘 increases when 𝑘 increases, it can also exist a value of 𝑚_𝑘 such that 𝑚_𝑘 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 𝐻₁=^∞_𝑘=0𝑓_𝑘(𝑥)𝑦^𝑘,or𝐻₂=^∞_𝑘=0𝑔_𝑘𝑦𝑥^𝑘. (16)

First we consider a general analytic system that we can always write into the following forms: 𝑥̇=𝑓(𝑥)+𝑦Φ₁𝑥,𝑦,𝑦̇=𝑔(𝑥)+𝑦Φ₂𝑥,𝑦, (17) or 𝑥̇=𝑓𝑦+𝑥Ψ₁𝑥,𝑦,𝑦̇=𝑔𝑦+𝑥Ψ₂𝑥,𝑦, (18) 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 series 𝐻₁ (𝐻₂, resp.) be a first integral of system (17) (system (18), resp.) among others, the first condition is 𝑓^′₀(𝑥)𝑓(𝑥)+𝑓₁(𝑥)𝑔(𝑥)=0, (𝑓^′₀(𝑦)𝑔(𝑦)+𝑓₁(𝑦)𝑓(𝑦)=0, resp.).

In order that this condition generates a collection of recursive differential equations where each 𝑓_𝑖(𝑥) does not depend on 𝑓_𝑖+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 𝑥̇=𝑃𝑥,𝑦=𝑝𝑥1+𝑃₀(𝑥)+_𝑗≥1𝑃_𝑗(𝑥)𝑦^𝑗,𝑦̇=𝑄𝑥,𝑦=𝑥𝑄₀(𝑥)−𝑞𝑦1+𝑄₁(𝑥)+_𝑗≥2𝑄_𝑗(𝑥)𝑦^𝑗, (19) where 𝑃_𝑗(𝑥)=∑_𝑖≥0𝑎_𝑖𝑗𝑥^𝑖 and 𝑄_𝑗(𝑥)=∑_𝑖≥0𝑏_𝑖𝑗𝑥^𝑖 are analytic functions, defined in a neighborhood of the origin.

Lemma 5. System (19) is orbitally equivalent to 𝑥̇=𝑃𝑥,𝑦=𝑝𝑥1+𝑃₀(𝑥)+_𝑗≥1𝑃_𝑗(𝑥)𝑦^𝑗,𝑦̇=𝑄𝑥,𝑦=−𝑞𝑦+_𝑗≥2𝑄_𝑗(𝑥)𝑦^𝑗, (20) that is, 𝑄₀(𝑥)≡𝑄₁(𝑥)≡0. Moreover, (𝑝,𝑞) are coprimes.

Proof. First we prove that we can achieve 𝑄₀(𝑥)≡0. Assume that 𝑄₀(𝑥)≢0 and let 𝑁=min{𝑖∈ℕ∪{0}∶𝑏_𝑖0≠0}. The change of variable 𝑢=𝑥, 𝑣=𝑦−(𝑏_𝑁0/(𝑝(𝑁+1)+𝑞))𝑥^𝑁+1 transforms system (19) into 𝑢̇=𝑝𝑢1+𝑃^∼₀(𝑢)+^∼_𝑗≥1𝑃_𝑗(𝑢)𝑣^𝑗,𝑣̇=𝑢𝑄^∼₀(𝑢)−𝑞𝑣1+𝑄^∼₁(𝑢)+^∼_𝑗≥2𝑄_𝑗(𝑢)𝑣^𝑗, (21) where 𝑄^∼₀(𝑢)=∑^∼_𝑖≥0𝑏_𝑖0𝑢^𝑖 and 𝑏^∼_𝑖0=0 for 𝑖=0,1,…,𝑁 because 𝑣̇=𝑦̇−((𝑁+1)𝑏_𝑁0/(𝑝(𝑁+1)+𝑞))𝑢^𝑁𝑥̇ hence 𝑢𝑄^∼₀(𝑢)=𝑏_𝑁0𝑢^𝑁+1−𝑞𝑏_𝑁0𝑝()𝑢𝑁+1+𝑞^𝑁+1−𝑝(𝑁+1)𝑏_𝑁0𝑝()𝑢𝑁+1+𝑞^𝑁+1+𝒪𝑢^𝑁+2=𝒪𝑢^𝑁+2. (22) Therefore, by means of successive change of variables we can transform system (19) into 𝑢̇=𝑝𝑢1+𝑃₀(𝑢)+_𝑗≥1𝑃_𝑗(𝑢)𝑦^𝑗,𝑣̇=−𝑞𝑣1+𝑄₁(𝑢)+_𝑗≥2𝑄_𝑗(𝑢)𝑣^𝑗. (23) To complete the proof it is enough to apply the scaling 𝑑𝑡=𝑑𝑇/(1+𝑄₁(𝑢)).

Lemma 6. Let 𝒢 be a formal function with 𝒢(0)=0. The system 𝑥̇𝑦̇=+𝒢𝑥𝑝𝑥−𝑞𝑦𝑝𝑥𝑞𝑦^𝑞𝑦^𝑝, (24) is formally integrable if and only if 𝒢≡0.

Proof. The sufficient condition is trivial since 𝑥^𝑞𝑦^𝑝 is a first integral of (𝑥̇,𝑦)̇^𝑇=(𝑝𝑥,−𝑞𝑦)^𝑇.
We now will prove the necessary condition. Let 𝐅(𝑥,𝑦)=(𝑝𝑥,−𝑞𝑦)^𝑇+(𝑝𝑥,𝑞𝑦)^𝑇𝒢(ℎ), where ℎ=𝑥^𝑞𝑦^𝑝 and 𝒢(ℎ)=∑_𝑗≥1𝑔_𝑗ℎ^𝑗. 𝒢≢0; otherwise, the proof is finished. We consider 𝑗₀=min{𝑗∈N∣𝑔_𝑗≠0}. If 𝐼 is a first integral of this system, then there exists 𝑀∈ℕ such that 𝐼=ℎ^𝑀+⋯ and 0=∇𝐼⋅𝐅=2𝑝𝑞𝑔_𝑗0ℎ^𝑗0+1+⋯. Therefore, 𝑔_𝑗0=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≥0𝑓_𝑗(𝑥)𝑦^𝑗 with 𝑓_𝑗(𝑥) a formal function is a first integral and moreover 𝑓₀=⋯𝑓_𝑝−1=0.

Proof. By applying the Poincare-Dulac normal form, there exists a change of variable to transform (20) into 𝑢̇𝑣̇=+𝑝𝑢∑−𝑞𝑣^𝑘≥1𝑎_𝑘𝑢ℎ^𝑘_𝑘∑_≥1𝑏_𝑘𝑣ℎ^𝑘, (25) where ℎ=𝑢^𝑞𝑣^𝑝. 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 functions ℱ(ℎ)=∑_𝑘≥1𝑐_𝑘ℎ^𝑘 and 𝒢^∼(ℎ)=∑_𝑘≥1𝑑_𝑘ℎ^𝑘 with 𝑐_𝑘=(𝑞𝑎_𝑘−𝑝𝑏_𝑘)/2𝑝𝑞 and 𝑑_𝑘=(𝑞𝑎_𝑘+𝑝𝑏_𝑘)/2𝑝𝑞, we get 𝑢̇𝑣̇=𝑢𝑝𝑢−𝑞𝑣1+ℱ^𝑞𝑣^𝑝+𝑝𝑢𝑞𝑣^∼𝒢𝑢^𝑞𝑣^𝑝, (26) 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 𝑢^′𝑣^′=+𝒢𝑢𝑝𝑢−𝑞𝑣𝑝𝑢𝑞𝑣^𝑞𝑣^𝑝, (27) where 𝒢=𝒢^∼/(1+ℱ) is a formal function with 𝒢(0)=0.
If this system is formally integrable, applying Lemma 6, we obtain 𝒢=0. Therefore, 𝐻(𝑥,𝑦)=(𝑥+⋯)^𝑞𝑦^𝑝(1+⋯)^𝑝=∑_𝑘≥𝑝𝑓_𝑘(𝑥)𝑦^𝑘 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≥𝑝𝑓_𝑘(𝑥)𝑦^𝑘 if for each 𝑘∈ℕ such that 𝑘=𝑀𝑝 with 𝑀∈ℕ is verified that the derivative 𝑔^(𝑀𝑞)_𝑘(0)=0, where 𝑔_𝑘 for 𝑘≥𝑝 is the analytical function𝑔_𝑘(𝑥)𝑔∶=^𝑘(𝑥)𝑝1+𝑃₀(𝑥)𝑓_𝑝(𝑥)^𝑘,𝑓_𝑝(𝑥)=exp−^𝑥₀𝑞𝑃₀(𝑠)𝑝𝑠1+𝑃₀(𝑠),𝑔𝑑𝑠_𝑝𝑔=0,𝑎𝑛𝑑𝑓𝑜𝑟𝑘>𝑝_𝑘(𝑥)=^𝑘−𝑝_𝑗=1𝑘−𝑗𝑓_𝑘−𝑗(𝑥)𝑄_𝑗+1(𝑥)+𝑓^′_𝑘−𝑗(𝑥)𝑃_𝑗(𝑥), (28) and for 𝑘≥𝑝𝑓_𝑘(𝑥)=𝑥^{𝑘𝑞/𝑝}𝑓_𝑝(𝑥)^𝑘𝐶_𝑘−^𝑥𝑠^{−1−(𝑘𝑞/𝑝)}𝑔_𝑘(𝑠)𝑑𝑠. (29)

Proof. By Proposition 7, we have 𝑓₀=⋯=𝑓_𝑝−1=0. If we impose that 𝐻(𝑥,𝑦) be a first integral of system (20), we obtain that the first condition is −𝑝𝑞𝑓_𝑝(𝑥)+𝑝𝑥𝑓^′_𝑝(𝑥)1+𝑃₀(𝑥)=0. (30) The next coefficients for each power of 𝑦 are of the form −𝑞𝑘𝑓_𝑘(𝑥)+𝑝𝑥1+𝑃₀(𝑥)𝑓^′_𝑘(𝑥)+𝑔_𝑘(𝑥)=0, (31) where 𝑔_𝑝=0 and 𝑔_𝑘(𝑥) for 𝑘>𝑝 depends of the previous 𝑓_𝑖(𝑥) for 𝑖=𝑝, 𝑝+1,…,𝑘−1. More specifically we obtain the expression (28).
The solution of homogeneous equation associated to (31) is 𝑓^ℎ_𝑘(𝑥)=exp𝑘𝑞𝑝𝑥1+𝑃₀(𝑥). (32) We can define the following analytical function 𝑏(𝑥)=−(𝑞/𝑝𝑥)(𝑃₀(𝑥)/(1+𝑃₀(𝑥))) and the integrand of the previous expression of 𝑓^ℎ_𝑘 is a rational function which admits the following fraction decomposition: 𝑘𝑞𝑝𝑥1+𝑃₀(𝑥)=𝑘𝑞(𝑥).𝑝𝑥+𝑘𝑏 (33) Therefore, 𝑓^ℎ_𝑘(𝑥)=𝑥^{𝑘𝑞/𝑝}exp(∫𝑘𝑏(𝑥))=𝑥^{𝑘𝑞/𝑝}(exp(∫^𝑥₀𝑏(𝑠)𝑑𝑠))^𝑘 and for the first equation which corresponds to 𝑘=𝑝 we have that 𝑔_𝑝(𝑥)=0. Therefore, 𝑓^ℎ_𝑝(𝑥) has the form 𝑓^ℎ_𝑝(𝑥)=𝐶_𝑝𝑥^𝑞(𝑓_𝑝(𝑥))^𝑝 where 𝑓_𝑝(𝑥)=exp(∫^𝑥₀𝑏(𝑠)𝑑𝑠) is an analytic function with 𝑓_𝑝(0)=1. So, 𝑓^ℎ_𝑘(𝑥)=𝐶_𝑘𝑥^{𝑘𝑞/𝑝}𝑓_𝑝(𝑥)^𝑘 and the solution of (31) is given by 𝑓_𝑘(𝑥)=𝑥^{𝑘𝑞/𝑝}𝑓_𝑝(𝑥)^𝑘𝐶_𝑘−^𝑥₀𝑠^{−𝑘𝑞/𝑝}𝑓_𝑝(𝑠)^−𝑘𝑝𝑠1+𝑃₀(𝑠)𝑔_𝑘(𝑠)𝑑𝑠. (34) Taking into account the form of 𝑔_𝑘(𝑥), we obtain the expression (29) for 𝑓_𝑘. 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 𝐶_𝑘=0, we have that each 𝑓_𝑘(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≥𝑝𝑓_𝑘(𝑥)𝑦^𝑘.

Corollary 9. System (20) admits a formal first integral of the form 𝐻(𝑥,𝑦)=∑_𝑗≥𝑝𝑓_𝑗(𝑥)𝑦^𝑗 if one of the following conditions holds: (a)𝑃₀(𝑥)≡0 and for each 𝑘=𝑀𝑝, 𝑀∈ℕ, 𝑔_𝑘(𝑥) is a polynomial of degree at most 𝑀𝑞−1; (b)𝑃₀(𝑥)≡𝑎𝑥^𝑟, with 𝑟 a positive integer and for each 𝑘=𝑀𝑝, 𝑀∈ℕ, 𝑔_𝑘(𝑥) is a rational function of the form 𝑔_𝑘(𝑥)=𝑔^∼_𝑘(𝑥)/(1+𝑎𝑥^𝑟)^{−1+𝑞/𝑟𝑝} where 𝑔^∼_𝑘 is a polynomial of degree at most 𝑀𝑞−1.

Proof. In case (a) it is enough to apply Theorem 8 for 𝑓_𝑝(𝑥)≡1 and 𝑔_𝑘(𝑥)=𝑔_𝑘(𝑥)/𝑝. For 𝑘=𝑀𝑝, 𝑀∈ℕ and we have 𝑔_𝑘^(𝑀𝑞)(𝑥)=𝑔_𝑘^(𝑀𝑞)/𝑝=0, because the degree of 𝑔_𝑘(𝑥) is less than 𝑞𝑀.
In case (b), we have that 𝑓_𝑝(𝑥)≡(1+𝑎𝑥^𝑟)^{−𝑞/𝑟𝑝}, and 𝑔_𝑘(𝑥)=𝑔_𝑘(𝑥)/𝑝(1+𝑎𝑥^𝑟)^{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≥𝑝𝑓_𝑗(𝑥)𝑦^𝑗 then the following conditions hold. (a)If for each 𝑘 such that 𝑘𝑞/𝑝∉ℕ, 𝑔_𝑘 is a polynomial then it is possible to choose 𝑓_𝑘 polynomial with deg(𝑓_𝑘)=deg(𝑔_𝑘). (b)If for each 𝑘=𝑀𝑝, 𝑀∈ℕ, 𝑔_𝑘(𝑥) is a polynomial with deg(𝑔_𝑘)≤𝑀𝑞−1 then it is possible to choose 𝑓_𝑘 polynomial with deg(𝑓_𝑘)≤𝑀𝑞−1.

Proof. By applying Theorem 8, we obtain 𝑓_𝑝(𝑥)≡1 and 𝑔_𝑘(𝑥)=𝑔_𝑘(𝑥)/𝑝. If we choose 𝐶_𝑘=0 in (29), we obtain 𝑓_𝑘(𝑥)1=−𝑝𝑥^{𝑘𝑞/𝑝𝑥}𝑠^{−1−𝑘𝑞/𝑝}𝑔_𝑘(𝑠)𝑑𝑠. (35) 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 𝑓_𝑘(𝑥)=−(1/𝑝)𝑥^𝑀𝑞 ∫^𝑥𝑠^{−1−𝑀𝑞}𝑔_𝑘(𝑠)𝑑𝑠. Therefore, we obtain the result applying that 𝑔_𝑘 is a polynomial with deg(𝑔_𝑘)≤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 𝑥̇=𝑝𝑥+_𝑗≥1𝑃_𝑗(𝑥)𝑦^𝑗,𝑥̇=−𝑞𝑦+_𝑗≥2𝑄_𝑗(𝑥)𝑦^𝑗, (36) where 𝑃_𝑗 and 𝑄_𝑗 are polynomials such that deg(𝑃_𝑗)≤⌊𝑗𝑞/𝑝⌋, deg(𝑄_𝑗)≤⌊(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 Proposition 7, if system (36) is integrable then a first integral is of the form 𝐻(𝑥,𝑦)=∑_𝑗≥𝑝𝑓_𝑗(𝑥)𝑦^𝑗 where 𝑓_𝑗(𝑥) is a formal function. Moreover, 𝑃₀(𝑥)≡0 and, therefore, by applying Theorem 8 we obtain 𝑓_𝑝(𝑥)≡1, 𝑔_𝑘(𝑥)=𝑔_𝑘(𝑥)/𝑝, 𝑔_𝑝=0 and 𝑓_𝑝(𝑥)=𝑥^𝑞.
We will prove now by induction that deg(𝑓_𝑗)≤⌊𝑗𝑞/𝑝⌋ and deg(𝑔_𝑗)≤⌊𝑗𝑞/𝑝⌋−1 for all 𝑗∈ℕ and 𝑝≤𝑗.
For 𝑗=𝑝 is true since 𝑓_𝑝(𝑥)=𝑥^𝑞 and 𝑔_𝑝=0. If we suppose that the hypothesis is true for 1≤𝑗≤𝑗₀−1, we have to prove that deg(𝑔_𝑗0)≤⌊𝑗₀𝑞/𝑝⌋−1 and deg(𝑓_𝑗0)≤⌊𝑗₀𝑞/𝑝⌋.
On the other hand, applying (28), we have deg𝑔_𝑗0≤max_{1≤𝑗≤𝑗0−𝑝}deg𝑓_𝑗0−𝑗+deg𝑄_𝑗+1,deg𝑓^′_𝑗0−𝑗+deg𝑃_𝑗≤𝑗₀𝑞𝑝−1, (37) since for 1≤𝑗≤𝑗₀−𝑝, we have deg𝑓_𝑗0−𝑗+deg𝑄_𝑗+1≤𝑗₀−𝑗𝑞/𝑝+𝑗𝑞𝑝≤𝑗−1₀𝑞𝑝𝑓−1,deg^′_𝑗0−𝑗+deg𝑃_𝑗≤𝑗₀−𝑗𝑞𝑝−1+𝑗𝑞𝑝≤𝑗₀𝑞𝑝−1. (38)
By applying Lemma 10, we have that deg(𝑓_𝑗0)≤⌊𝑗₀𝑞/𝑝⌋−1. In particular, deg(𝑓_𝑗0)≤⌊𝑗₀𝑞/𝑝⌋.
Therefore, if 𝑘=𝑀𝑝, deg(𝑔_𝑘)≤𝑀𝑞−1, 𝑔_𝑘^𝑀𝑞(0)=0 and 𝑔_𝑘^𝑀𝑞(0)=0.
To finish the proof, it is enough to apply Theorem 8.

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 𝑥^𝑞𝑦^𝑝 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 series 𝐻₁ 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 𝐹₁𝑥,𝑦=^∞_𝑖+𝑗=2𝑎_𝑖𝑗𝑥^𝑖𝑦^𝑗,𝐹₂𝑥,𝑦=^∞_𝑖+𝑗=2𝑏_𝑖𝑗𝑥^𝑖𝑦^𝑗, (39) where 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 are arbitrary constants. If we impose that 𝐻₁ be a first integral of system (5), we obtain that the first condition is ^∞_𝑖=2𝑏_𝑖0𝑥^𝑖𝑓₁(𝑥)+𝑝𝑥𝑓^′₀(𝑥)+^∞_𝑖=2𝑎_𝑖0𝑥^𝑖𝑓^′₀(𝑥)=0. (40)

The proof of this proposition is also straightforward developing in powers of 𝑦 and taking the coefficient of the power 𝑦⁰. Hence in order to have a recursive differential equations where each 𝑓_𝑖(𝑥) does not depend on 𝑓_𝑖+1(𝑥), we must impose that 𝑏_𝑖0=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 𝑏_𝑖0=0 for all 𝑖 the system (5) has 𝑦 as invariant algebraic curve.

The next coefficients for each power of 𝑦 are of the form 𝑘−𝑞+^∞_𝑖=1𝑏_𝑖1𝑥^𝑖𝑓_𝑘(𝑥)+𝑝𝑥+^∞_𝑖=2𝑎_𝑖0𝑥^𝑖𝑓^′_𝑘(𝑥)+𝑔_𝑘(𝑥)=0, (41) where 𝑔_𝑘(𝑥) depends of the previous 𝑓_𝑖(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 𝑓₁(𝑥)=exp−−𝑞+∑^∞_𝑖=1𝑏_𝑖1𝑥^𝑖𝑝𝑥+∑^∞_𝑖=2𝑎_𝑖0𝑥^𝑖. (42) The integrand of the previous expression of 𝑓₁ is a rational function which admits the following fraction decomposition: −𝑞+∑^∞_𝑖=1𝑏_𝑖1𝑥^𝑖𝑝𝑥+∑^∞_𝑖=2𝑎_𝑖0𝑥^𝑖=𝐴𝑥+𝐵(𝑥)𝑝+∑^∞_𝑖=2𝑎_𝑖0𝑥^𝑖−1, (43) where 𝐴=−𝑞/𝑝 and 𝐵(𝑥) is a formal series. Therefore, 𝑓₁(𝑥)=𝑥^𝑞/𝑝𝑓₁(𝑥) where 𝑓₁ 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, 𝑏_𝑖0=0 for 𝑖≥2 and 𝑏_𝑖1=0 for 𝑖≥1 and 𝑔_𝑘(𝑥)=^∼𝑔_𝑘(𝑥)𝑃(𝑥,0)𝑥𝑓₁(𝑥)^−𝑘with𝑔^∼_𝑘^(𝑘𝑞)(0)=0. (44)

Proof. Under the assumptions of the theorem, we have that 𝑓₀=0, 𝑓₁=𝑥^𝑞𝑓₁(𝑥), and (41) takes the form −𝑘𝑞𝑓_𝑘(𝑥)+𝑃(𝑥,0)𝑓^′_𝑘(𝑥)+𝑔_𝑘(𝑥)=0, (45) whose solution is given by 𝑓_𝑘(𝑥)=𝑥^𝑘𝑞𝑓₁(𝑥)^𝑘𝐶_𝑘−^𝑥𝑠^−𝑘𝑞𝑓₁(𝑠)^−𝑘𝑃()𝑔𝑠,0^𝑘(𝑠)𝑑𝑠. (46) Taking into account the form of 𝑔_𝑘(𝑥), we obtain 𝑓_𝑘(𝑥)=𝑥^𝑘𝑞𝑓₁(𝑥)^𝑘𝐶_𝑘−^𝑥𝑠^{∼−1−𝑘𝑞}𝑔_𝑘(𝑠)𝑑𝑠. (47) 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 𝑓_𝑘(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 Proposition 13 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 𝑔_𝑘(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, 𝑎_𝑖0=𝑏_𝑖0=0 for 𝑖≥2 and 𝑏_𝑖1=0 for 𝑖≥1 and 𝑔_𝑘(𝑥) is a polynomial of degree at most 𝑘𝑞−1; (b)𝑝=1, 𝑎_𝑖0=0 for 𝑖≠𝑟+1, 𝑏_𝑖0=0 for 𝑖≥2 and 𝑏_𝑖1=0 for 𝑖≥1 with 𝑟 a positive integer and 𝑔_𝑘(𝑥) is a rational function of the form 𝑔_𝑘(𝑥)=𝑔^∼_𝑘(𝑥)/(1+𝑎𝑥^𝑟)^{−1+𝑘𝑞/𝑟} where 𝑔^∼_𝑘 is a polynomial of degree at most 𝑘𝑞−1.

Proof. In case (a) for we have 𝑓₀=0 and 𝑓₁=𝑥^𝑞 (41) take the form −𝑘𝑞𝑓_𝑘(𝑥)+𝑥𝑓^′_𝑘(𝑥)+𝑔_𝑘(𝑥)=0, (48) whose solution is given by 𝑓_𝑘(𝑥)=𝐶_𝑘𝑥^𝑘𝑞−𝑥^𝑘𝑞𝑥𝑠^{−1−𝑘𝑞}𝑔_𝑘(𝑠)𝑑𝑠, (49) and taking into account that 𝑔_𝑘(𝑥), that depends on 𝑓_𝑖 for 𝑖=1,2,…,𝑘−1, is at most of degree 𝑘𝑞−1, we obtain that 𝑓_𝑘 is a polynomial. Moreover, in this case system (5) becomes 𝑥̇=𝑝𝑥+𝑦𝜙₁,̇𝑥,𝑦𝑦=−𝑞𝑦+𝑦𝜙₂𝑥,𝑦, (50) where 𝜙₁ and 𝜙₂ are analytic functions.
In case (b), we have that 𝑓₀=0, 𝑓₁=𝑥^𝑞(1+𝑎𝑥^𝑟)^−𝑞/𝑟 and (41) takes the form −𝑘𝑞𝑓_𝑘(𝑥)+𝑥𝑓^′_𝑘(𝑥)+𝑎𝑥^𝑟+1𝑓^′_𝑘(𝑥)+𝑔_𝑘(𝑥)=0, (51) whose solution is given by 𝑓_𝑘(𝑥)=𝑥^𝑘𝑞1+𝑎𝑥^{𝑟−𝑘𝑞/𝑟}×𝐶_𝑘−^𝑥𝑠^{−1−𝑘𝑞}(1+𝑎𝑠)^{−1+𝑘𝑞/𝑟}𝑔_𝑘(𝑠)𝑑𝑠, (52) and taking into account the form of 𝑔_𝑘(𝑥), we obtain that 𝑓_𝑘(𝑥) is also of the form 𝑓_𝑘(𝑥)=𝑝_𝑘(𝑥)/(1+𝑎𝑥^𝑟)^{−𝑘𝑞/𝑠} where 𝑝_𝑘(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. Corollary 14 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 form 𝐻₁ 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 the 1∶−1 resonant quadratic system given by 𝑥̇=𝑥+𝑎₂₀𝑥²+𝑎₁₁𝑥𝑦+𝑎₀₂𝑦²,𝑦̇=−𝑦+𝑏₂₀𝑥²+𝑏₁₁𝑥𝑦+𝑏₀₂𝑦². (53) 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 𝑥̇=𝑥+𝑎₁₁𝑥𝑦+𝑎₀₂𝑦²,𝑦̇=−𝑦+𝑏₀₂𝑦² (54) 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 𝐻=𝑏𝑥,𝑦₀₂𝑦−1^{−1−(𝑎11/𝑏02)}×2𝑎₀₂−2𝑎₀₂𝑎₁₁+𝑎₀₂𝑏₀₂𝑦+𝑎³₁₁−𝑎₁₁𝑏²₀₂𝑥𝑦+𝑎₀₂𝑎²₁₁+𝑎₀₂𝑎₁₁𝑏₀₂𝑦². (55)

Example 16. Consider the 1 : −1 resonant Lotka–Volterra planar complex quartic system given by 𝑥̇=𝑥1−𝑎₁₀𝑥−𝑎₀₁𝑦−𝑎₂₀𝑥²−𝑎₁₁𝑥𝑦−𝑎₀₂𝑦²−𝑎₃₀𝑥³−𝑎₂₁𝑥²𝑦−𝑎₁₂𝑥𝑦²−𝑎₀₃𝑦³,𝑦̇=−𝑦1−𝑏₁₀𝑥−𝑏₀₁𝑦−𝑏₂₀𝑥²−𝑏₁₁𝑥𝑦−𝑏₀₂𝑦²−𝑏₃₀𝑥³−𝑏₂₁𝑥²𝑦−𝑏₁₂𝑥𝑦²−𝑏₀₃𝑦³. (56) 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 𝑓_𝑖(𝑥) is given by −𝑖𝑓_𝑖(𝑥)+𝑥𝑓^′_𝑖(𝑥)+𝑔_𝑖(𝑥)=0 where 𝑔_𝑖(𝑥) is(𝑖−3)𝑏₀₃𝑓_𝑖−3(𝑥)+(𝑖−2)𝑏₀₂𝑓_𝑖−2(𝑥)+(𝑖−2)𝑏₁₂𝑥𝑓_𝑖−2(𝑥)+(𝑖−1)𝑏₀₁𝑓_𝑖−1(𝑥)+(𝑖−1)𝑏₁₁𝑥𝑓_𝑖−1(𝑥)+(𝑖−1)𝑏₂₁𝑥²𝑓_𝑖−1(𝑥)+𝑖𝑏₁₀𝑥𝑓_𝑖(𝑥)+𝑖𝑏₂₀𝑥²𝑓_𝑖(𝑥)+𝑖𝑏₃₀𝑥³𝑓_𝑖(𝑥)−𝑎₀₃𝑥𝑓^′_𝑖−3(𝑥)−𝑎₀₂𝑥𝑓^′_𝑖−2(𝑥)−𝑎₁₂𝑥²𝑓^′_𝑖−2(𝑥)−𝑎₀₁𝑥𝑓^′_𝑖−1(𝑥)−𝑎₁₁𝑥²𝑓^′_𝑖−1(𝑥)−𝑎₁₀𝑥²𝑓^′_𝑖(𝑥)−𝑎₂₀𝑥³𝑓^′_𝑖(𝑥)−𝑎₃₀𝑥⁴𝑓^′_𝑖(𝑥). (57) Now we impose that 𝑔_𝑖(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 −𝑖𝑓_𝑖(𝑥)+𝑥𝑓^′_𝑖(𝑥)+𝑔_𝑖(𝑥)=0 where 𝑔_𝑖(𝑥) is given by (𝑖−3)𝑏₀₃𝑓_𝑖−3(𝑥)+(𝑖−2)𝑏₀₂+𝑏₁₂𝑥𝑓_𝑖−2(𝑥)+(𝑖−1)𝑏₀₁𝑓_𝑖−1(𝑥)−𝑎₀₃𝑥𝑓^′_𝑖−3(𝑥)−𝑎₀₂𝑥𝑓^′_𝑖−2(𝑥)−𝑎₁₂𝑥²𝑓^′_𝑖−2(𝑥)−𝑎₀₁𝑥𝑓^′_𝑖−1(𝑥). (58) We obtain that its solution is given by 𝑓_𝑖(𝑥)=𝐶_𝑖𝑥^𝑖−𝑥^𝑖𝑥𝑠^−1−𝑖𝑔_𝑖(𝑠)𝑑𝑠, (59) and taking into account that 𝑔_𝑖(𝑥) is of degree ≤𝑖−1, we can establish the following result.

Proposition 17. The system 𝑥̇=𝑥1−𝑎₀₁𝑦−𝑎₁₂𝑥𝑦²−𝑎₀₃𝑦³,𝑦̇=−𝑦1−𝑏₀₁𝑦−𝑏₀₂𝑦²−𝑏₁₂𝑥𝑦²−𝑏₀₃𝑦³ (60) has an analytic first integral in a neighborhood of the origin.

In fact, Proposition 17 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]).

Acknowledgments

A. Algaba and C. García are supported by a MICINN/FEDER Grant no. MTM2010-20907-C02-02 and by the Consejería de Educacíon y Ciencia de la Junta de Andalucía (projects EXC/2008/FQM-872, TIC-130, and FQM-276). J. Giné is partially supported by a MICINN/FEDER Grant no. MTM2011-22877 and by Generalitat de Catalunya Grant no. 2009SGR 381.