Conditions for the existence of a center for the Kukles homogeneous systems

In this work, we study necessary and sufficient conditions for the existence of centers in two families of linear centers with homogeneous quartic and quintic nonlinearities. Systems of this class are called Kukles homogeneous systems. Systems of this type were studied, for the first time, by Kukles, who studied a linear center with cubic nonhomogeneous nonlinearities.