Centers of cubic polynomial differential systems

An equilibrium point $ p $ of a differential system in the plane $ {\mathbb{R}}^2 $ is a center if there exists a neighborhood $ U $ of $ p $ such that $ U\setminus \{p\} $ is filled with periodic orbits. A difficult classical problem in the qualitative theory of differential systems in the plane is the problem of distinguishing between a focus and a center. In this paper we characterize when the origin of coordinates is a center of the following cubic polynomial differential systems$ \begin{equation*} \dot{x} = -y, \qquad \dot{y} = x + a_{1} x^{2}+ a_{2} x y+ a_{3} y^{2}+ A, \end{equation*} $where $ A $ is an arbitrary nonzero monomial of degree 3. Moreover we provide all topologically different phase portraits when $ A = a_{4}x^{3} $.