Centers and foci for a class of cubic systems (Q5951121)

Necessary and sufficient criteria are established for the equilibrium \(O(0,0)\) to be a center of the following cubic system \[ dx/dt=x+Ax^2+Bx^3+4Cy^3,\quad dy/dt=-y+Msy/3+Nx^2y/3.\tag{1} \] The author proves the existence of such a system of the form (1) for which \(O(0,0)\) is a focus of order 12. One can find some interesting references related to the present problem. At the end of this paper, the author relates an open problem for further research.