CENTER CONDITIONS FOR A CUBIC DIFFERENTIAL SYSTEM WITH ONE INVARIANT STRAIGHT LINE AND ONE INVARIANT CUBIC

In this paper the conditions for the existence of one invariant straight line and one invariant cubic in a cubic differential system with a singular point of a center or a focus type, when these curves have not intersecting points, are found. It is proved that the singular point is a center if and only if the first five Lyapunov quantities vanish. The center conditions were determined by using the method of Darboux integrability.