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

In the present work for the system u x = y(1+Dx+Px 2 ), u y = −x+Ax 2 + 3Bxy+Cy 2 +Kx 3 +3Lx 2 y+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 u x = yP0(x), u y = −x+P2(x)y 2 +P3(x)y 3 , for which 35 cases of a center are shown. We prove the existence of systems of the form u x = y(1+Dx+Px 2 ), u y = −x+�y +Ax 2 +Cy 2 +Kx 3 + 3Lx 2 y +Mxy 2 +Ny 3 with eight limit cycles in the neighborhood of the origin of coordinates.