Quadratic differential systems with complex conjugate invariant lines meeting at a finite point

Abstract

In this article we study the class QS2cIL of quadratic differential systems with two complex conjugate invariant lines meeting at a finite point. From the literature we know that quadratic systems with invariant lines of total multiplicity at least four or with the line at infinity filled up with singularities are integrable via the method of Darboux and hence they have no limit cycles. These could only occur if we have only the two complex lines, and the line at infinity, all simple. We first find all integrable systems in QS2cIL due to the presence of invariant lines. We next indicate a gap in the 1986 proof of Suo and Chen that systems in QS2cIL have at most one limit cycle and we give a complete proof of this result. Finally we give the topological classification of QS2cIL yielding 22 phase portraits three of which with a limit cycle.