Variety of the center and limit cycles of a cubic system, which is reduced to lienard form

In the present work for the system $\dot{x}=y(1+Dx+Px^2)$, $\dot{y}=-x+Ax^2+3Bxy+Cy^2+Kx^3+3Lx^2y+Mxy^2+Ny^3$ 25 cases are given when the point $O(0,0)$ is a center. We also consider a system of the form $\dot{x}=yP_0(x)$, $\dot{y}=-x+P_2(x)y^2+P_3(x)y^3$, for which 35 cases of a center are shown. We prove the existence of systems of the form $\dot{x}=y(1+Dx+Px^2)$, $\dot{y}=-x+\lambda y +Ax^2+Cy^2+Kx^3+3Lx^2y+Mxy^2+Ny^3$ with eight limit cycles in the neighborhood of the origin of coordinates.