Solution of the Center Problem for a Class of Polynomial Differential Systems

Consider the polynomial differential system of degree m of the form \(\eqalign{&\dot{x}=-y(1+\mu(a_{2}x-a_{1}y))+x(\nu(a_{1}x+a_{2}y)+\Omega_{m-1}(x,y)),\cr &\dot{y}=x(1+\mu(a_{2}x-a_{1}y))+y(\nu(a_{1}x+a_{2}y)+\Omega_{m-1}(x,y)),}\) $$\begin{array}{cc}& \dot{x}=-y(1+\mu (a_{2}x-a_{1}y))+x(\nu (a_{1}x+a_{2}y)+{\mathrm{\Omega }}_{m-1}(x,y)),\\ & \dot{y}=x(1+\mu (a_{2}x-a_{1}y))+y(\nu (a_{1}x+a_{2}y)+{\mathrm{\Omega }}_{m-1}(x,y)),\end{array}$$ where μ and ν are real numbers such that \((\mu^{2}+\nu^{2})(\mu+\nu(m-2))(a_{1}^{2}+a_{2}^{2})\ne 0,m > 2\) $$({\mu }^2+{\nu }^2)(\mu +\nu (m-2))(a_1^2+a_2^2)\ne 0,m>2$$ and Ω_m−1(x,y) is a homogenous polynomial of degree m − 1. A conjecture, stated in J. Differential Equations 2019, suggests that when ν = 1, this differential system has a weak center at the origin if and only if after a convenient linear change of variable (x,y) → (X,Y) the system is invariant under the transformation (X,Y,t) → (−X,Y, −t). For every degree m we prove the extension of this conjecture to any value of ν except for a finite set of values of μ.