The family of cubic differential systems with two real and two complex distinct infinite singularities and invariant straight lines of the type ( 3 , 1 , 1 , 1 )

Abstract

In this article we consider the class CSL 7 2 r 2 c ∞ of non-degenerate real planar cubic vector fields, which possess two real and two complex distinct infinite singularities and invariant straight lines of total multiplicity 7, including the line at infinity. The classification according to the configurations of invariant lines of systems possessing invariant straight lines was given in articles published from 2014 up to 2022. We continue our investigation for the family CSL 7 2 r 2 c ∞ possessing configurations of invariant lines of type ( 3 , 1 , 1 , 1 ) and prove that there are exactly 42 distinct configurations of this type. Moreover we construct all the orbit representatives of the systems in this class with respect to affine group of transformations and a time rescaling.