In this paper we prove the Darboux integrability of a cubic differential system with a singular point of a center typer having at least two parallel invariant straight lines.

Abstract. For cubic differential systems with two parallel invariant straight lines and at least one invariant conic it is proved that a singular point with pure imaginary eigenvalues (a weak focus) is a centre if and only if the first three Liapunov quantities Lj, j = 1, 2, 3 vanish. Mathematics Subject Classification (2000). Primary 34C05; Secondary 58F14.

No abstract

No abstract

We find conditions for a singular point O(0, 0) of a center or a focus type to be a center, in a cubic differential system with one irreducible invariant conic. The presence of a center at O(0, 0) is proved by constructing integrating factors.