Sufficient $GL(2, \mathbb{R})$-invariant center conditions for some classes of two-dimensional cubic differential systems

The autonomous two-dimensional polynomial cubic systems of differential equations with pure imaginary eigenvalues of the Jacobian matrix at the singular point $(0,0)$ are considered in this paper. The center problem was studied for three classes of such systems: the class of cubic systems with zero divergence of the cubic homogeneities ($S_3\equiv 0$), the class of cubic systems with zero divergence of the quadratic homogeneities ($S_2\equiv 0$) and the class of cubic systems with nonzero divergence of the quadratic homogeneities ($S_2\not\equiv 0$). For these systems, sufficient $GL(2, \mathbb{R})$-invariant center conditions for the origin of coordinates of the phase plane were established.