An inverse approach to the center problem

We consider analytic or polynomial vector fields of the form X=-y+X∂∂x+x+Y∂∂y,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}=\left( -y+X\right) \dfrac{\partial }{\partial x}+\left( x+Y\right) \dfrac{\partial }{\partial y},$$\end{document} where X=X(x,y))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X=X(x,y))$$\end{document} and Y=Y(x,y))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y=Y(x,y))$$\end{document} start at least with terms of second order. It is well-known that X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document} has a center at the origin if and only if X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document} has a Liapunov–Poincaré local analytic first integral of the form H=12(x2+y2)+∑j=3∞Hj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=\dfrac{1}{2}(x^2+y^2)+\sum _{j=3}^ {\infty } H_j$$\end{document}, where Hj=Hj(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_j=H_j(x,y)$$\end{document} is a homogenous polynomial of degree j. The classical center-focus problem already studied by Poincaré consists in distinguishing when the origin of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document} is either a center or a focus. In this paper we study the inverse center problem, i.e. for a given analytic function H of the previous form defined in a neighborhood of the origin, we determine the analytic or polynomial vector field X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document} for which H is a first integral. Moreover, given an analytic function V=1+∑j=1∞Vj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=1+\sum _{j=1}^ {\infty } V_j$$\end{document} in a neighborhood of the origin, where Vj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_j$$\end{document} is a homogenous polynomial of degree j, we determine the analytic or polynomial vector field X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document} for which V is a Reeb inverse integrating factor. We study the particular case of centers which have a local analytic first integral of the form H=12(x2+y2)1+∑j=1∞Υj,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H=\dfrac{1}{2}(x^2+y^2)\,\left( 1+ \sum _{j=1}^{\infty } \Upsilon _j\right) , $$\end{document} in a neighborhood of the origin, where Υj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Upsilon _j$$\end{document} is a homogenous polynomial of degree j for j≥1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\ge 1.$$\end{document} These centers are called weak centers, they contain the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We have characterized the expression of an analytic or polynomial differential system having a weak center at the origin We extended to analytic or polynomial differential systems the weak conditions of a center given by Alwash and Lloyd for linear centers with homogeneous polynomial nonlinearities. Furthermore the centers satisfying these weak conditions are weak centers.


Abstract
We consider analytic or polynomial vector fields of the form X = (−y + X ) where X = X (x, y)) and Y = Y (x, y)) start at least with terms of second order.
It is well-known that X has a center at the origin if and only if X has a Liapunov-Poincaré local analytic first integral of the form H = 1 2 (x 2 + y 2 ) + ∞ j=3 H j , where H j = H j (x, y) is a homogenous polynomial of degree j. The classical center-focus problem already studied by Poincaré consists in distinguishing when the origin of X is either a center or a focus. In this paper we study the inverse center problem, i.e. for a given analytic function H of the previous form defined in a neighborhood of the origin, we determine the analytic or polynomial vector field X for which H is a first integral. Moreover, given an analytic function V = 1+ ∞ j=1 V j in a neighborhood of the origin, where V j is a homogenous polynomial of degree j, we determine the analytic or polynomial vector field X for which V is a Reeb inverse integrating factor. We study the particular case of centers which have a local analytic first integral of the form H = 1 2 (x 2 + y 2 ) 1 + ∞ j=1 ϒ j , in a neighborhood of the origin, where ϒ j is a homogenous polynomial of degree j for j ≥ 1. These centers are called weak centers, they contain the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We  an analytic or polynomial differential system having a weak center at the origin We extended 1 Introduction be the real planar analytic or polynomial vector field associated to the real planar differential systemẋ = P(x, y),ẏ = Q(x, y), where the dot denotes derivative with respect to an independent variable t.
In what follows we assume that The study of the centers of analytical or polynomial differential systems (2) has a long history. The first works are due to Poincaré [1,2] and Dulac [3]. Later on were developed by Bendixson [4], Frommer [5], Liapunov [6] and many others.
In the paper [7] we study the polynomial vector fields where X and Y are homogenous polynomials of degree m. In the present paper we mainly extend the results of [7] to systems (3) when X and Y are analytic or polynomial functions whose Taylor expansions at the origin do not contain constant and linear terms. System (3) always has a center or a focus at the origin. The classical center-focus problem already studied by Poincaré consists in distinguishing when the origin of system (3) is either a center or a focus, i.e, asks about conditions on the coefficients of X and Y under which O is a center or focus.
In the study of the center-focus problem the following theorems play a very important role (see for instance [1,2,6,8]) Theorem 1 For the analytic differential system (3) there exists a formal power series W = ∞ n=2 W n := 1 2 (x 2 + y 2 ) + ∞ n=3 W n (x, y), where W j = W j (x, y) is a homogenous polynomial of degree j such that dW dt = ∞ j=1 v j (x 2 + y 2 ) j+1 , where v j are the Poincaré-Liapunov constants.
Assume that the formal power series W converges. If the constants v j = 0 for j ∈ N then there exists a first integral H := 1 2 (x 2 + y 2 ) + ∞ j=3 W j , and consequently the origin is a center. If there exists a first non-zero Liapunov constant v j , then the origin is a stable focus if v j < 0 and unstable if v j > 0. Poincaré and Liapunov proved the next two results, see for instance [1,2,6,9,10].
Theorem 2 A planar polynomial differential systeṁ of degree m has a center at the origin if and only if it has a first integral of the form where X j , Y j and H j are homogenous polynomials of degree j.
The analytic function (5) is called the Poincaré-Liapunov local first integral.
Theorem 3 An analytic planar differential systeṁ has a center at the origin if and only if it has a first integral of the form (5).
Theorem 2 is due to Poincaré, and Theorem 3 is due to Liapunov. From Theorems 1, 2 and 3 it is clear that an analytic or polynomial differential system (3) has a center at the origin if and only if the Poincaré-Liapunov constants v k = 0 for k ≥ 1 (Poincaré's criterion). Moreover, the v k 's are polynomials over Q in the coefficients of the polynomial differential system. A necessary and sufficient condition to have a center is then the annihilation of all these constants. In view of the Hilbert's basis theorem this occurs if and only if for a finite number of k, k < j and j sufficiently large, v k = 0. Unfortunately, trying to solve the center problem computing the Poincaré-Liapunov constants is in general not possible due to the huge computations.
Although we have an algorithm for computing the Poincaré-Liapunov constants for linear type center, we have no algorithm to determine how many of them need to be zero to imply that all of them are zero for cubic or higher degree polynomial differential systems. Bautin [11] showed in 1939 that for a quadratic polynomial differential system, to annihilate all v k 's it suffices to have v k = 0 for i = 1, 2, 3. So the problem of the center is solved for quadratic systems. This problem was solved for the cubic differential systems with homogenous nonlinearities (see for instance [8,12,13]).
We recall the following definition. Let U be an open and dense set in R 2 . We say that a non-constant C r with r ≥ 1 function F : U → R is a first integral of the analytic or polynomial vector field X on U , if F(x(t), y(t)) is constant for all values of t for which the solution (x(t), y(t)) of X is defined on U . Clearly F is a first integral of X on U if and only if X F = 0 on U . Now we shall introduce another criterion for solving the center problem due to Reeb. We need the following definitions and notions. A function V = V (x, y) is an inverse inverse integrating factor of system (2) in an open subset We note that {V = 0} is formed by orbits of system (2). The function 1/V defines an inverse integrating factor in U \{V = 0} of system (2) which allows to compute a first integral for (2) in U \{V = 0}.
We consider now the relation between the existence of a center and that of an inverse integrating factor for analytic or polynomial vector fields. The main result is given by the following theorem see [14].
Theorem 4 (Reeb 's criterion) The analytic differential system (6) has a center at the origin if and only if there is a local nonzero analytic inverse integrating factor of the form V = 1 + h.o.t. in a neighborhood of the origin.
An analytic inverse integrating factor having the Taylor expansion at the origin V = 1 + h.o.t. is called a Reeb inverse integrating factor.
Darboux gave his geometric method of integration in his seminal work [15] of 1878. The geometric method of Darboux uses algebraic invariant curves of a polynomial differential system for computing a first integral of the system. There were numerous publications on the problem of the center using the Darboux method during the last part of the 20th century and the beginning of the 21st century (see for instance [16][17][18]). In fact there is the following conjecture due to Zoladek [18], see also [16]. See these papers for more details on this conjecture.

Conjecture 5
Suppose that the polynomial differential system (3) has a center at the origin. Then this system has a Darboux first integral or an algebraic symmetry.
To show that a singular point is a center for system (3) we have two basic mechanisms: we either apply Poincaré-Liapunov Theorem and we show that we have a local analytic first integral, or we apply the Reeb inverse integrating factor. Another mechanism for detecting centers has been given by Mikonenko see [19].
The main objective of the present paper is to analyze the center problem from the inverse point of view (see for instance [7,20,21]).
We state and solve the following inverse problems for the centers of analytic and polynomial vector fields.

Problem 6 Inverse Poincaré-Liapunov's Problem Determine the analytic (polynomial) planar vector fields
for which the given function (5) is a local analytic first integral where X j = X j (x, y), Y j = Y j (x, y) for j ≥ 2 are homogenous polynomials of degree j.

Problem 7 Inverse Reeb Problem
Determine the analytic (polynomial) planar vector fields (7) for which the V = 1 + ∞ j=1 V j where V j = V j (x, y) for j ≥ 2 are homogenous polynomials of degree j, is the Reeb inverse integrating factor, i.e.
Hence, either given an analytic function H of the form (5) we shall determine the analytic functions X and Y in (3) in such a way that the function H is a first integral of the differential system (3), or given an analytic function 1 + ∞ j=1 V j in a neighborhood of the origin we shall determine the analytic functions X and Y in (3) in such a way that the analytic differential system (3) has the function V as a Reeb inverse integrating factor.
The solutions of the given inverse problem for analytic case is given in Theorem 16 and for the polynomial case is given in Theorem 17.
The solution of the problems 6 and 7 provide the expression for the analytic or polynomial functions X and Y , i.e. we determine all the homogeneous parts of X and Y which are given by is the Poisson bracket of the functions f and g, and is an arbitrary homogenous polynomial of degree i. For the Inverse Poincaré-Liapunov's Problem the polynomials g i are arbitrary functions such that 1 + ∞ i=1 g i converges in the neighborhood of the origin, and for the Inverse Reeb Problem we obtain where the homogeneous polynomial F i of degree i > 2 is arbitrary, F 2 = (x 2 + y 2 )/2, and 1 + ∞ i=1 F i converges in the neighborhood of the origin. From the solutions of the inverse problem 6 and 7 we obtain the next results.
Corollary 8 An analytic differential system (6) has a center at the origin if and only if either it has a first integral H of the form (5), or it has an inverse Reeb integrating factor V . Moreover differential system (6) can be written asẋ = V {H, x},ẏ = V {H, y}.
We find a new class of centers which we call weak centers. We say that a center at the origin of an analytic differential system is a weak center if in a neighborhood of the origin it has an analytic first integral of the form H = where ϒ j is a homogenous polynomial of degree j. We have characterized the expression of an analytic or polynomial differential system having a weak center at the origin. We prove that the following statements are equivalent.
(a) If an analytic (or polynomial) differential systems has a weak center at the origin then it can be written asẋ where = (x, y) and = (x, y) are convenient analytic (polynomial) functions (see Theorem 20). (b) Letż = i z + R(z,z) be the system (4) in complex coordinates z = x + iy andz = x − iy. Then if this system has a weak center at the origin then R(z,z) = z (z,z) (see Proposition 19). This is a very simple criterium to determine the non existence of a weak center. (c) If an analytic (or polynomial) differential system (4) (or (6)) has a weak center, then the first integral (5) satisfies that its homogenous polynomial H j for j = 3, . . . , k ≤ ∞ is Moreover we prove that the uniform isochronous centers and the isochronous holomorphic centers are weak centers.
It is well known the following result (see for instance [22]). Let X be an analytic vector field associated to differential system (3). Then X has either a focus or a center at the origin, and under a formal change of coordinates the differential system associated to X can be written into the Birkhoff normal forṁ where S j = S j (x 2 + y 2 ) for j = 1, 2 are formal series in the variable x 2 + y 2 . Clearly these differential equations are particular case of (10).
We have extended the weak conditions of a center given by Alwash and Lloyd in [23] for linear centers with homogenous polynomial nonlinearities (see Proposition 14), to a general analytic or polynomial differential system see Theorem 32. Furthermore the centers satisfying the generalized weak conditions of a center, introduced in Theorem 32, are weak centers.
Finally we observe that the given above inverse Problem 6 was solve for polynomial vector fields with homogenous nonlinearities in [7].

Preliminary concepts and results
In the proofs of the results that we provide in this paper it plays an important role the following results. Proof Indeed, if we change x = cos t, y = sin t then it is easy to show that

Proposition 10 The next relation holds
Hence, 2π 0 {H 2 , }| x=cos t, y=sin t dt = (cos t, sin t)| t=2π t=0 = 0. The following result is due to Liapunov (see Theorem 1, page 276 of [6] In particular, for n = 2 the partial differential equation has a unique solution V if and only if As a simple consequence of Theorem 11 we have the next result. In what follows some examples of planar vector fields having a center are studied.

Hamiltonian system
When system (3) is Hamiltonian, i.e. there exists a function F = F(x, y) such that is a first integral.

Reversible system
Besides Hamiltonian systems there is another class of systems (3) for which the origin is a center, namely the reversible systems satisfying the following definition. We say that system (3) is reversible with respect to the straight line l through the origin if it is invariant with respect to reversion about l and a reversion of time t (see for instance [24]).
The following criterion goes back to Poincaré see for instance [25], p.122.

Theorem 13 The origin of system (3) is a center if the system is reversible.
In particular this theorem is applied for the case when (3) is invariant under the transfor-

Weak condition for a center
The following condition weak condition for a center was due to Alwash and Lloyd [7,23], see also [7].

Proposition 14
The origin is a center of a polynomial differential system of the forṁ where X m and Y m are homogenous polynomial of degree m, if there exists μ ∈ R such that and either m = 2k is even; or m = 2k − 1 is odd and μ = 2k; or m = 2k − 1 is odd, μ = 2k and In [26] the author proved that if μ = 2m then system (12) has the rational first integral

Cauchy-Riemann condition for a center
Another particular case of differential systems with a center are the systems satisfing the Cauchy-Riemann conditions (see for instance [24]).

Proposition 15 (Cauchy-Riemann condition for a center) Let O be a center of (2). Then O is isochronous center if P and Q satisfy the Cauchy-Riemann equations
A center of system (3) for which (13) holds is called a holomorphic center, which is also an isochronous center, see for more details [27] and [28]. We recall that a center of system (3) located at the origin is an isochronous center if all the periodic solutions in a neighborhood of the origin have the same period.

Statement of the main results
The main results are stated in the following four subsections.

Analytic and polynomial vector fields with a linear type center
The inverse Poincaré-Liapunov's problem (see Problem 6) and inverse Reeb problem (see Problem 7) for the analytic (k = ∞) planar vector fields has been solved in the following theorem which provides the expressions of the analytic differential systems (6) in function of its first integral (5) or in function of its Reeb inverse integrating factor.

Theorem 16
Consider the analytic vector field X associated to differential system (6). Then this vector field has a Poincaré-Liapunov local first integral if and only if it has a Reeb inverse integrating factor. Moreover, (i) the analytic differential system (6)

is a local first
integral can be written aṡ where g j = g j (x, y) is an arbitrary homogenous polynomial of degree j which we choose in such a way that the series ∞ j=1 g j converge in the neighborhood of the origin. (ii) The differential system (4) for which V = 1 + ∞ j=1 V j is a Reeb integrating factor can be written asẋ is an arbitrary homogenous polynomial of degree j which we choose in such a way that ∞ j=2 F j converges, i.e. F is an arbitrary Poincaré-Liapunov local first integral.
The inverse Poincaré-Liapunov's problem and inverse Reeb problem for the polynomial planar vector fields (k = m < ∞) has been solved in the following theorem which provides the expressions of the analytic differential systems (6) in function of its first integral (5) or in function of its Reeb integrating factor.

Theorem 17
Consider the polynomial vector field X associated to differential system (4).

Then this polynomial vector field has a Poincaré-Liapunov local first integral if and only if
it has a Reeb inverse integrating factor. Moreover, the differential system associated to the vector field X for which H = (x 2 + y 2 )/2 + h.o.t. is a local first integral can be written aṡ where , γ is an oriented curve (see for instance [29]), τ j = τ j (x, y) is a convenient analytic function in the neighborhood of the origin such that τ j (0, 0) = 1, and g j = g j (x, y) is an arbitrary homogenous polynomial of degree j which we choose in such a way that 1/ is the inverse Reeb inverse integrating factor which satisfies the first order partial differential equation Remark 18 From the proof of Theorem 17 it follows that (17) is equivalent to the infinite number of first order partial differential equations with unknowns the homogenous polynomials g j of degree j ≥ m. Hence by Proposition 10 we obtain the conditions The first condition, by Corollary 12 guarantees the existence of the solution g m of first equation of (18), the second condition, again by Corollary 12, guarantees the existence of the solution g m+1 of the second equation of (18), and so on.

Analytic and polynomial vector fields with local analytic first integral of the
We say that a differential system (3) has a weak center at the origin if it has a local analytic The aim of this section is to study the weak centers for analytic and polynomial differential systems.
In the study of the weak centers plays a fundamental role the differential systems of the formẋ where = (x, y) and = (x, y) are convenient analytic (polynomial) functions. This differential system is called − differential system Proposition 19 Consider a differential system of the forṁ where R = R(z,z) is an analytic (or polynomial) function at the origin, z = x + i y and z = x − i y. Then this system can be rewritten as (19) if and only if R(z,z) = z (z,z) where is analytic (or polynomial). (20) it follows thaṫ hence by comparing with (19) we get that = V and = U. The reciprocity it is easy to obtain. Indeed, system (19) can be written asż = i z + z( + i ).
Theorem 20 An analytic differential system (6) has a weak center at the origin if and only if this system can be written aṡ where ϒ 0 = 1, g 0 = 1, g j and ϒ j are homogenous polynomials of degree j for j ≥ 1 and for j ≥ m + 1, we obtain necessary and sufficient conditions under which the polynomial differential system (21) of degree m and has the first integral where μ j = μ j (x, y) is a convenient analytic function in the neighborhood of the origin for j = 1, . . . , m − 1.

Corollary 21
Define a homogenous polynomial of degree j − 1, then differential system (21) and conditions (22) can be rewritten as followṡ for j > m respectively. Moreover the following relation holds.

Corollary 22
The center of polynomial differential system is a weak center if and only if Proof Follows trivially from the previous theorem in view of formula (16).
The singular point of system (3) located at the origin is an isochronous center if all the periodic solutions in a neighborhood of it has the same period.

Corollary 23
The weak center of a polynomial differential system (19) is an isochronous center if and only if where (r, θ) are the polar coordinates, and r satisfies that is a constant on any periodic solution surrounding the isochronous center.
holds for a nonzero constant κ; or equivalently in polar coordinates (r, θ) such that x = r cos θ, y = r sin θ , we have thatθ = κ.
Corollary 24 An analytic differential system (19) has a uniform isochronous center at the origin if and only iḟ and

Moreover polynomial differential system of degree m (19) has a uniform isochronous center at the origin if and only if (25) holds and
Example 25 For polynomial differential system (12) we have that (25) becomeṡ This system has the Poincaré-Liapunov first integral , (see for a proof Proposition 12 of [7]) The inverse approach to study the uniform isochronous center was given in [30]. Theorem 20 has the following additional corollary.

Corollary 26
Assume that the planar differential system (4) has a center at the origin. Then this center is a holomorphic isochronous center if and only if system (4) can be written as (19), i.e. is a weak center, with the function and satisfying the Cauchy-Riemann conditions Moreover, a polynomial differential system (19) with a holomorphic center at the origin is Darboux integrable. Here we provide the explicit expression of a such first integral.
We observe that the Darboux integrability of polynomial differential systems with a holomorphic center at the origin is a well known result (see for instance [31]).

Remark 27
From Corollaries 26 and 24 it follows that all the uniform isochronous centers and all the holomorphic isochronous centers for polynomial differential systems are always weak centers.
It is important to observe that there is not a relation between isochronous centers and weak centers, i.e. there exist isochronous centers which are not weak centers and weak centers which are not isochronous centers. Then for instance the quadratic isochronous centeṙ is not a weak center because it has the first integral H = (9 − 24y + 32x 2 ) 2 /(3 − 16y) for more details see [32]. On the other hand the quadratic systeṁ has a weak center at the origin because it has the first integral H = (1 + 2y)(x 2 + y 2 ) but it is not isochronous see [32]. In fact in [7] we provide all the quadratic system with weak centers.
Now we introduce the following definitions and notations. Let R[x, y] be the ring of all real polynomials in the variables x and y, and let X be the polynomial vector field (2) of degree m. Let g = g(x, y) ∈ R[x, y]\R. Then g = 0 is a polynomial of degree at most m − 1, which is called the cofactor of g = 0. A function g = g(x, y) satisfying that g = 0 is an invariant curve (i.e. formed by orbits of the vector field X ) is called partial integral. If g ∈ R[x, y]\R then g is called a polynomial partial integral or a Darboux polynomial. If the polynomial g is irreducible in R[x, y], then we say that the invariant algebraic curve g = 0 is irreducible, and that its degree is the degree of the polynomial g. A first integral F of the polynomial vector field (1) (x, y) . . . g λ r r (x, y), where k, h, g 1 , . . . , g r are polynomials and λ 1 , . . . , λ r are complex constants. For more details on the so-called Darboux theory of integrability see for instance Chapter 8 of [33].
We introduce the following definition. We say that a polynomial vector field X of degree m is quasi-Darboux integrable if there exist r polynomial partial integrals g 1 , . . . , g r and s non-polynomial C r with r > 0 partial integrals f 1 , . . . , f s satisfying y) is a convenient polynomials of degree m − 1, for j = 1, . . . , s such that the function is a first integral, where k = k(x, y), h = h(x, y) are polynomials, and λ 1 , . . . , λ r , κ 1 , . . . , κ s , are complex constants. We observe that a generalization of the Darboux theory was developed in the paper [34], which evidently contains the above definition with another name, but for our aim we shall use the name of quasi-Darboux integrable.
We have the following conjecture.

Conjecture 28 A polynomial differential system (19) having a weak center at the origin is quasi-Darboux integrable.
This conjecture is supported by several facts which we give below. , y), where H 2 = 0 is an invariant algebraic curve and f = 0 is an analytic (non polynomial) invariant curve with cofactor 2 and −2 respectively.

Proposition 29 A polynomial differential system (19) with a weak center at the origin is quasi-Darboux integrable in a neighborhood of the origin with the first integral H
Note that the first integral provided in Proposition 29 is only defined in a neighborhood of the origin, while a quasi-Darboux first integral as it is given in (26) is defined in a dense set of R 2 .

Center problem for analytic or polynomial vector fields with a generalized weak condition of a center
First we prove the following two propositions.

Proposition 30
Assume that a differential system (3) satisfies the relation with μ ∈ R\{0}. Then the system can be written as in (19) with and = (x, y) an arbitrary analytic function in a neighborhood of the origin. Moreover system (3) has the inverse integrating factor (x 2 + y 2 ) μ/2 , and it can be written aṡ Note that if in (27) we have that μ = 0, then system (3) is a Hamiltonian system.

Proposition 31
Consider the polynomial differential system (2) of degree m which satisfy the relations Then there exist polynomialsH = m+1 j=3 H j and G = m−1 j=1 G j of degree m + 1 and m − 1 respectively such that system (2) can be written aṡ Note that we have extended the definition of "weak condition for a center" given in subsection 2.3 for a quasi-homogenous polynomial differential system to a general analytic differential system. Proposition 14 can be generalized as follows.
Theorem 32 [Generalized weak condition of a center of an analytic (polynomial) differential systems] We consider an analytic (polynomial) differential system (6). Then the origin is a weak center if there exists μ ∈ R\{0} such that the following relations hold where μ ∈ R\{0}. Moreover this differential system can be written aṡ with λ = 2/μ and ϒ = ϒ(x, y) and q = q(H 2 ) are a convenient analytic functions.
If differential system (33) is a polynomial differential system of degree m, i.e. ϒ = ϒ(x, y) is a polynomials of degree m − 1 and q(H 2 )) =

. then the system (33) is quasi Darboux-integrable with the first integral F which is given in what follows
The algebraic curves H 2 = 0 and
The algebraic curve H 2 = 0 and non-polynomial curve The algebraic curves H 2 = 0 and

The given first integrals has the following Taylor extension at the origin F = H 2 (1 + h.o.t.)
Consequently the origin is a weak center.
In an analogous way we can study the analytic case.

Linear centers with degenerate infinity
We shall study the following class of differential systemṡ where R m−1 = R m−1 (x, y) is a convenient nonzero homogenous polynomial of degree m − 1. Such system are polynomial differential systems with a degenerate infinity. This name is due to the fact that in the Poincaré compactification of (38) the line at infinity is filled with singular points.

Proposition 33 Assume that a polynomial differential system (4) has a center at the origin with a first integral H given in (5). Then this system has a degenerate infinity if it can be written asẋ
Proposition 33 characterizes the polynomial differential systems having a degenerate infinity and a linear type center at the origin.

Proposition 34 Polynomial differential system (19) has a degenerate infinity if it can be written asẋ
i.e. m−1 = 0.
All the results of this subsection are proved in section 7.

The proofs of Sect. 3.1
Proof of Theorem 16 First we prove the "only if part". Assume that the analytic differential system (6) has a Poincaré-Liapunov local first integral. Then we shall see that it can be written as (14), where 1 + ∞ j=1 g j is the Reeb inverse integrating factor. Following exactly the ideas of the proof of Theorem 9 of [7] we have where g n = g n (x, y) is an arbitrary homogenous polynomial of degree n. Hence, since ∞ j=1 g j converges in a neighborhood of the origin, we get thaṫ Note that the function 1 + ∞ j=1 g j is an analytic integrating factor of the differential system (14) i.e. it is a Reeb inverse integrating factor. Now we prove the " if " part. We assume that system (4) has a Reeb inverse integrating factor. From the equation (8) From the first equation of (41) we get that y) is an arbitrary homogenous polynomial of degree 2. From the second equation of (41) we obtain where F 3 = F 3 (x, y) is an arbitrary homogenous polynomial of degree 3. From the third equation of (41) we obtain where F 4 = F 4 x, y) is an arbitrary homogenous polynomial of degree 4. By continuing this process we get y) and F j = F j (x, y) are homogenous polynomial in the variables x and y of degree j,we get thatX where k > 1 and F k = F k (x, y) is an arbitrary homogenous polynomial of degree j, for j ≥ 3, such that the series F = ∞ j=2 F j , converges at neighborhood of the origin. By considering that we are interesting in studying the linear type center then X 1 = −y and X 1 = x then we have that F 2 = (x 2 + y 2 )/2. Therefore is a Poincaré-Liapunov local first integral this prove the " if " part of the theorem.
Moreover, by summing we geṫ Thus the proof of the theorem follows.
Proof of Corollary 8 Follows trivially from the proof of Theorem 16 (see statement (i) and (ii)).

Remark 35
From the "only if" part follows that the arbitrariness which we determine the vector fields with the given Poincaré-Liapunov local first integral is related with the Reebs inverse integrating factor V = 1 + ∞ j=2 g j and from the "if" part follows that the arbitrariness which we determine the vector fields with the given Reebs inverse integrating factor is related with the Poincaré-Liapunov local first integral F = (x 2 + y 2 )/2 + F 3 + F 4 + · · · .
Proof of Theorem 17 Now we assume that the vector field X is polynomial of degree m. First we prove the "only if" part. From (40) it follows that if X n = Y n = 0 for n > m, theṅ Clearly, if X n = Y n = 0 for n ≥ m + 1, then for k ≥ 1. This system of partial differential equations of first order is compatible if and only if the following relations hold Clearly that these conditions always hold if m + k − 1 is odd, and consequently in this case there exists a unique homogenous polynomial of degree m + k − 1. We shall study partial differential equations (46) under the conditions (47). For k = 1 from (46) we get where g m = g m (x, y) is an arbitrary homogenous polynomial of degree m which satisfies the first order partial differential equations (see first equation of (47)) Hence the two first partial differential system (46) are compatible, consequently integrating the 1-form (48) we obtain On the other hand from (46) and using that H j are homogenous polynomial of degree j, we get that For k = 2 system (46) becomes which in view of (48) system (49) can be written as where g m+1 = g m+1 (x, y) is an arbitrary homogenous polynomial of degree m + 1 which satisfies the first order partial differential equation Hence, in view of Proposition 12 we get that On the other hand, from (50) and in view of the fact that H j are homogenous polynomial of degree j we get that Under the condition (51) system (50) is compatible, consequently after the integration de 1-form we get that which in view of (48) system (49) can be written as where g m+2 = g m+2 (x, y) is an arbitrary homogenous polynomial of degree m + 2 which satisfies the first order partial differential equation Under this condition system (52) is compatible, consequently after the integration of the 1-form and using the property of homogenous polynomial we get that By continuing this process we deduce that where β j = β j (x, y) are homogenous polynomial of degree j, and g m+k−2 is an arbitrary homogenous polynomial of degree m + k − 2 which we choose as a solution of the first order partial differential equation Thus for k ≥ 5, where α k is a convenient homogenous polynomial of degree k. From these results it follows that the homogenous polynomials H j+1 and g j−1 for j > m we determine by the line integral and as a solution of the linear partial differential equation respectively. y) is a convenient analytical function, for j = 2, . . . , m + 1. Hence, if ∞ j=3 g j converges in a neighborhood of the origin, then in view of the Taylor expansion

By summing we finally obtain
· · · , we get that Therefore the function (55) can be written as follows On the other hand, by summing (47), (51), (53), (54) and etc. we get Hence we obtain that the polynomial differential system (45) of degree m can be written as (15) where 1 + ∞ j=2 g j is the Reeb inverse integrating factor. In short the proof of the "only if part" and the statement (i) follows. This proves the "only if part" of the theorem. Now we prove the "if" part. We assume that V = 1 + ∞ j=2 V j is the Reeb inverse integrating factor. From (43) and (44) it follows that If X j = Y j = 0 for j ≥ m + 1, then for k ≥ m + 1. System of partial differential equations of first order (56) is compatible if and only if where k ≥ m + 1. The proof of statement (ii) can be obtained analogously to the proof of statement (i), if we take g j = V j and H j+1 = F j+1 for j = 1, . . . , m.
Finally we observe that from (16) it follows that

From the condition
, we get the condition (17). In short the theorem is proved.
Proof of Corollary 9 It follows easily from the proof of Theorem 17.
In order to illustrate Theorem 17 we study the following polynomial systems.

Example 36
We shall determine the quadratic system having the Reeb integrating factor where A and b are nonzero constants. The quadratic polynomial differential system (42) in this case becomes a homogenous polynomial of degree 3, which satisfies the conditions (57) for k = 3. Hence we obtain that F 3 = bx 2 y + κ y 3 , where κ is a constant. Therefore Example 37 The quartic differential systems with homogenous nonlinearitieṡ can be written as (15) The homogenous polynomials H 5 and g 3 are From the condition (53) for m = 4 and with g 1 = g 2 = 0 and H 3 = H 4 = 0 we get that the origin is a focus of (58) because 2π 0 {H 5 , g 3 }| x=cos t,y=sin t dt = 0.

The proofs of Sect. 3.2
Proof of Theorem 20 Necessity We suppose that system (3) has a weak center at the origin. Consequently there exists an analytic local first integral H = H 2 (1 + ∞ j=1 ϒ j ) := H 2 . Then from Theorem 16 it follows the necessary and sufficient conditions on the existence of a linear type center for an analytic differential system differential system. Thus (14) becomes Sufficiency Now we suppose that (21) holds and show that then the origin is a weak center. Indeed, from (59) we obtain thatḢ 2 then H = H 2 is a first integral, consequently the origin is a weak center.
The second statement is proved as follows. Under the assumption which is equivalent to the equations for j > m + 1, from (59) we get the following polynomial differential equations of degree m.ẋ Here ϒ j is a convenient homogenous polynomial of degree j, such that H 2 = (x 2 + y 2 )/2, H j+2 = H 2 ϒ j , for j = 1 . . . m + 1 and g j is an arbitrary homogenous polynomial of degree j satisfying (17). Consequently in view of (16) we obtain the local first integral (23). Thus the theorem is proved Proof of Corollary 21 From the relations H j = H 2 ϒ j−1 and it follows that Inserting H j+1 into the relations j = {H j+1 , H 2 } + g 1 {H j , H 2 } + · · · + g j−2 {H 3 , H 2 }, we obtain that here we use the relations Hence after some computations we get that Then the proof of the corollary follows easily.
In order to illustrate Theorem 20 we study the following polynomial systems.

Example 38
The following cubic polynomial differential system has a center at the origin (see [35]) Consequently this system can be rewritten as (19) with the functions and determined as follows 1 + = 1 2 (1 + (y + 1) 2 , = x − y − y 2 . and hence the center is a weak center.
Example 39 For the quadratic differential systeṁ the functions H 3 and g 1 are It is easy to show that the solution of equation (17) for m = 2, i.e. the equation Consequently from (16) the quadratic system has the following first integral and from it we obtain the Poincaré-Liapunov first integral H = In particular if d − A − b = 0, then the quadratic polynomial differential system has a weak center at the origin with H 3 = 2b y H 2 , and Poincaré-Liapunov first integral H = (1 + Ay) 2b/A−1 H 2 .
Proof of Corollary 23 First we observe that differential equations (19) in polar coordinates x = r cos θ, y = r sin θ becomeṡ r = r (r cos θ, r sin θ),θ = 1 + (r cos θ, r sin θ), hence in view of that the center is weak center, then the polar coordinates must be such that H (r cos θ, r sin θ) = C = constant. Hence we get that the weak center is an isochronous center if and only if (24) holds, thus the corollary is proved.

Proof of Corollary 24
It trivially follows from Corollary 21 assuming that j = 0 for j = 1, . . . , ∞. Hence, if we assume that {H j , g 1 } + · · · + {H 2 , g j−1 } = 0 for j > m + 1 then we obtain the conditions under which the polynomial differential system has a uniform isochronous center at the origin. Thus the proof of the corollary follows.

Proof of Corollary 26 From the Cauchy-Riemann conditions it is easy to obtain condition
After some computations we obtain the following expression for the first integral where z 0 = 0. In view of the relation and by considering condition (62) after tedious computations we obtain that the first integral is a Darboux first integral: So the first integral F has a Taylor expansion in the neighborhood of the origin has the form . Thus the holomorphic isochronous center is a weak center. In short the proposition is proved.
We observe that the problem on the existence the first integral for the complex differential system was study in particular in [31] Proof of Proposition 29 Since at the origin of system (19) there is a weak center, we have an analytic first integral H = H 2 f in a neighborhood of the origin. So clearly H 2 = 0 and f = 0 are invariant curves of system (19) y) is an arbitrary function. Denoting ν = 1 + we get that differential equations (3) coincide with (19). On the other hand in view of the relations which is equivalent to i.e. H λ 2 is inverse integrating factor. Thus differential system (3) can be written as (28) with F given by the formula (29). In short corollary is proved.
Proof of Proposition 31 Suppose that P and Q can be written as in (31) whereH and G are polynomials, and we shall see that such polynomials exist when (30) holds. Then Hence by considering that By considering that (30) holds, then in view of Corollary 12 we deduce that there exists a polynomial G = m−1 j=1 G j such that We can determine the functionH as follows , y) is the solution of equation (64). In short the proposition is proved.
We observe that from (31) it follows that thus in view of Proposition 10 we obtain that 2π 0 (x P(x, y) + y Q(x, y)) | x=cos t, y=sin t dt = 0, We consider an analytic differential system (6) under the assumptions (32). The previous result can be extended for the analytic vector field. Thus we have the following proposition.
Proposition 40 Let (2) be an analytic differential system which satisfies the relation 2π 0 ∂ P ∂ x + ∂ Q ∂ y | x=cos t, y=sin t dt = 0. Then there exist analytic functionsH = ∞ j=2 H j and G = ∞ j=1 G j such thaṫ Proof It is analogous to the proof of Proposition 31.
Finally We have the following remarks related with differential system (19).

The proofs of Sect. 3.3
Proof of Theorem 32 We shall study only the case when the differential system is a polynomial differential systems of degree m. It is possible to show that condition (27) is equivalent to (63). Hence from the first of condition of (32) and in view of Proposition 30 we get that a polynomial differential system (3) can be written as (28) with F given in the formula (29).
(c) The second of condition (32) for analytic differential system is necessary as it follows from the next example. If it holds then in view of Proposition 31 we get that analytic differential system (6) can be written as (31). In this case we have 2π 0 ∂(−y + X ) ∂ x + ∂(x + Y ) ∂ y | x=cos t, y=sin t dt = 2π 0 {H 2 , G} | x=cos t, y=sin t dt = 0, and that differential system (31) satisfies the second conditions of (32) for arbitrary G andH . Clearly there exist analytic function G andH for which the origin is a focus and the first conditions of (32) does not hold.

The proofs of Sect. 3.4
Proof of Proposition 33 From (15) and (38) it follows that ∂ H m+1 ∂ y + g 1 ∂ H m ∂ y + · · · + g m−1 y = −x R m−1 , Thus Hence by considering that H j is homogenous polynomial of degree j we get that Substituting this polynomial into (68) we obtain