The radical of the focal values ideal of the complex Kukles system

In the article it is considered the center-focus problem for complex Kukles system $\dot{x}=y, ~\dot{y}=-x+Ax^{2}+3Bxy+Cy^{2}+Kx^{3}+3Lx^{2}y+Mxy^{2}+Ny^{3}$. The problem is solved by the new method, obtained by A. P. Sadovski and based on the method of normal forms. Instead of investigating the variety of ideal of focal values it is proposed to study the variety of ideal with the basis – polynomials obtained by a new method. The study of the radical of such ideal is divided into two parts: the trivial case where $BN=0$, and the case of $BN\neq 0$. If $BN=0$ it is obtained five series of center conditions for the complex system, in particular, four series of center conditions for the real system. In the case of $BN\neq 0$ it can be assumed $B=N$ in the Kukles system. This assumption simplifies the further study of this case (it is obtained three series of the existence of the complex center, in particular, two series of the existence of the real center). Thus, as a result of research in the present paper it is presented the necessary and sufficient conditions for the complex and real centers existence for the complex and real Kukles systems respectively.