CUBIC SYSTEMS WITH SEVEN INVARIANT STRAIGHT LINES ALONG ONE DIRECTION, WHEN SOME STRAIGHT LINES ARE COMPLEX CONJUGATE

Abstract

Consider the generic cubic differential system x = P(x, y), y = Q(x, y), where P,Q ∈ R[x, y], max degP, degQ = 3, GCD P,Q = 1. If this system has enough invariant straight lines considered with their multiplicities, then, according to [1], we can construct a Darboux first integral. In this paper we obtain 8 canonical forms for cubic differential systems which possess invariant straight lines along one direction of total multiplicity seven including the straight line at the infinity and at least one planar invariant straight line is complex.