The center problem for the class of Λ-Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda -\varOmega $$\end{document} differential systems

The center problem, i.e. distinguish between a focus and a center, is a classical problem in the qualitative theory of planar differential equations which go back to Darboux, Poincaré and Liapunov. Here we solve the center problem for the class of planar analytic or polynomial differential systems x˙=-y+X=-y+∑j=2kXj,y˙=x+Y=x+∑j=2kYj,k≤∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{x}=-y+X=-y+\displaystyle \sum _{j=2}^k\,X_j,\quad \dot{y}=x+Y=x+\displaystyle \sum _{j=2}^k\,Y_j,\quad k\le \infty , \end{aligned}$$\end{document}where Xj=Xj(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_j=X_j(x,y)$$\end{document} and Yj=Yj(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_j=Y_j(x,y)$$\end{document} are homogenous polynomials of degree j>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j>1$$\end{document}, under the condition (x2+y2)∂X∂x+∂Y∂y=μxX+yYwithμ∈R\{0}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (x^2+y^2)\left( \dfrac{\partial X}{\partial x}+\dfrac{\partial Y}{\partial y} \right) =\mu \left( xX+yY\right) \quad \text{ with }\quad \mu \in \mathbb {R}\setminus \{0\}. \end{aligned}$$\end{document}Moreover we prove that these centers are weak centers, and additionally we provide their first integrals.


Introduction and main results
Let X = (−y + X) x + (x + Y) y be the real planar analytic or polynomial vector field associated to the real planar differential system where X j = X j (x, y) and Y j = Y j (x, y) are homogenous polynomials of degree j. The Poincaré center-focus problem asks about conditions on the coefficients of X and Y under which all trajectories of system (1) contained in a small open neighborhood of the origin are closed, of course with exception of the origin. One of the mechanism to solve the centerfocus problem is the following result due to Poincaré and Liapunov.

Theorem 1 A planar analytic or polynomial differential system (1) has a center at the origin if and only if it has a first integral of the form
where H j are homogenous polynomials of degree j.

The first integral H is called a Poincaré-Liapunov first integral.
A center is called a weak center if the Poincaré-Liapunov first integral can be written in the form The next theorem was proved in [5]. (3) is called a − differential equation. Another well-know mechanism to solve the center-focus problem is the Reeb criterium.

Theorem 2 A center of an analytic (polynomial) differential system (1) is a weak center if and only if it can be written in the form
Theorem 3 (Reeb's criterion, see for instance [9]). The analytic differential system (1) has a center at the origin if and only if there is a local analytic inverse integrating factor of the form V = 1 + ∑ ∞ j=1 g j (x, y), called in what follows the Reeb inverse integrating factor, in a neighborhood of the origin, where g j = g j (x, y) is a homogenous polynomial of degree j > 0.
The following result is proved in [8].
Corollary 1 Assuming that differential system (1) has a center at the origin. Then the

Poincaré-Liapunov first integral H can be written as
where T k is a homogenous polynomial for k ≥ 1 such that for k ≥ 1 where T 1 = g 1 , and V is the Reeb inverse integrating factor.
In particular if the vector field X is polynomial of degree m then (4) becomes Moreover if the center is a weak center then where 0 = 1.
A partial integral of the vector field X is a differentiable function G ∶ D ⟶ ℝ where D is an open subset of ℝ 2 satisfying with K a function of the same class than G.
We say that an analytic (polynomial) vector field X is quasi-Darboux integrable if there exist r polynomial partial integrals g 1 , … , g r ∈ ℝ[x, y] and s C r with r > 0 nonpolynomial partial integrals f 1 , … , f s being their cofactors K j = K j (x, y) analytic (polynomial) functions for j = 1, … , s such that the function is a first integral, where k = k(x, y), h = h(x, y) are analytic (polynomial) functions, and 1 , … , r , 1 , … , s , are complex constants. We remark that when the functions f j and K j for j = 1, … , s , k and h are polynomials the first integral F is called a generalized Darboux first integral (see for instance [2]), we also remark that the generalized Darboux first integrals are a particular class of the Liouvillian first integrals (see for more details [10]).
In [5] the following conjecture was stated.

Conjecture 1 Any analytic (polynomial) vector field with a weak center at the origin is
quasi-Darboux integrable.
This conjecture is supported in particular by the results given in the proof of the next Theorem 5 and the following two results. Note that the first result is a local one.

Theorem 4 Any analytic (polynomial) vector field with a weak center at the origin is locally quasi-Darboux integrable.
Theorem 4 is proved in Sect. 2 The weak conditions provided by Alwash and Lloyd [1] give sufficient conditions for the existence a center. More precisely, they proved: The main objective of this paper is to extend and improve the result of Proposition 1 to analytic and polynomial differential systems. Thus our two main results are the following ones. For the analytic systems one of the following conditions holds For the polynomial systems of degree m one of the following conditions holds (iv) = 2k ∈ {3, … , m + 1} and 2k−1 = 0.
Moreover differential system (1) satisfying (7) and having a weak center at the origin is quasi-Darboux integrable.

Corollary 2 Differential system (5) under condition (6) has a weak center at the origin if and only if
Theorem 5 and Corollary 2 are proved in Sect. 3.

Preliminary results
In the proofs of our results it plays an important role the following propositions and corollaries.
The Poisson bracket of two functions f and g is defined as

Proposition 2
The next relation holds The following corollary is due to Liapunov (see Theorem 1, page 276 of [4]).

Corollary 3
Let U = U(x, y) be a homogenous polynomial of degree m. The linear partial differential equation U(x, y)| x=cos t, y=sin t dt = 0.

Proof of Theorem 4
Indeed, if the analytic vector field has a weak center at the origin then it admits a Poincaré-Liapunov local analytic first integral at the origin y). From Theorem 2 we get that Ḣ 2 = 2H 2 (x, y), i.e. H 2 is a partial integral with analytic cofactor 2 (x, y). It is easy to show that the analytic function (x, y) is an analytic partial integral with cofactor −2 (x, y). In short the theorem is proved. ◻ In the proof of Theorem 5 and Corollary 2 we need the following two propositions which also appeared in the Ph.D. [8], and for completeness we prove them here.
Proof Indeed, if (11) holds then there exist a function G such that consequently (10) holds. Assume that (10) holds then and by considering that x y we get that Hence in view of the relation ∫ 2 0 H 2 , G | x=cos t, y=sin t dt = 0 we obtain that (11) holds. In short the proposition is proved. Now we prove formula (12) for differential system (9) under the condition (11). If (11) holds then from (10) we get that So Thus Consequently, by introducing the function = F + 2H 2 G, we get that Thus formula (12) is proved. In short the proposition is proved. ◻ Proposition 4 The analytic differential system (2) satisfying condition (7) can be written as the − differential system Where = (x, y) is a solution of the first order partial differential equation which in polar coordinates x = r cos , y = r sin becomes y).
Proof Indeed, from (7)  where is given by formula (15), and is a solution of equation (14). Thus the proposition is proved. ◻ A center O of system (1) is a uniform isochronous center if the equality ẋy − ẏx = (x 2 + y 2 ) holds for a nonzero constant ; or equivalently in polar coordinates (r, ) such that x = r cos , y = r sin , we have that ̇= . Clearly that from Theorem 2 it follows that uniform isochronous centers are weak center. Example 1 For differential system (5) under the condition (6) we get that differential system (13) and condition (14) becomes respectively. Consequently if − m − 1 ≠ 0 , then system (16) can be written as and if − m − 1 = 0 , then Since the differential system (17) in polar coordinates writes we get that the weak center in this case is a uniform center.
Therefore is a first integral, which in cartesian coordinates becomes Therefore is an analytic first integral defined in a neighborhood of the origin, i.e. it is a Poincaré-Liapunov first integral. From the expression of this first integral we obtain that the origin is a weak center. Clearly that H 2 and are partial integrals with analytic cofactor 2 and ( − 2) respectively. Now assume that ∈ ℕ�{2}. Let j = − 2, then −2 ( ) = −2 =constant. From (26) and in view of (24) we obtain that or equivalently the function f is We have that where ∶= (x, y) is a polynomial of degree − 1 and −1 is a constant such that This first integral in cartesian coordinates becomes Therefore from (20) we obtain that Thus in order to obtain a center at the origin from (20) we need that −1 = 0. From (24) we get that −2 is a constant, i.e. (2 ), sin(2 )) −F(cos(0), sin(0)) = −1 arctan(tan(2 )) 2 ∕2−1 . we have a center, by considering that is a center of a − equation we obtain that the origin is a weak center. Evidently that are partial integrals with analytic cofactor 2 and ( − 2) . If ≠ 2k then H is a first integral an analytic at the origin. Thus the origin is a weak center. Clearly that H 2 and + (x, y) are partial integrals with analytic cofactor 2 and ( − 2) . Finally we consider the case = 2. Then from (26) and (24) we have that The first integral H is such that lim (x,y)⟶ (0,0)H = 0 and it is not analytic. So we have a center, by considering that is a center of the − equation we obtain that the origin is a weak center. If = 2 then we get the following Poincaré-Liapunov first integral In short the corollary is proved. ◻

Example
In [6,7] we state the following conjecture  The center problem for the case when ( + m − 2)(a 2 1 + a 2 2 ) = 0, we solve in the next proposition.