Global Nilpotent Reversible Centers with Cubic Nonlinearities Symmetric with Respect to the x-Axis

Let \(P_3(x,y)\) and \(Q_3(x,y)\) be polynomials of degree three without constant or linear terms. We characterize the global centers of all polynomial differential systems of the form \(\dot{x} = y+ P_3(x,y)\), \(\dot{y} =Q_3(x,y)\) that are reversible and invariant with respect to the x-axis.