Classification of cubic differential systems with invariant straight lines of total multiplicity eight and two distinct infinite singularities

Abstract

In this article we prove a classification theorem (Main theorem) of real planar cubic vector fields which possess two distinct infinite singularities (real or complex) and eight invariant straight lines, including the line at infinity and including their multiplicities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of invariant polynomials. The algebraic invariants and comitants allow one to verify for any given real cubic system with two infinite distinct singularities whether or not it has invariant lines of total multiplicity eight, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems in this class, given in any normal form.

Keywords

• cubic vector fields
• configuration of invariant lines
• infinite and finite singularities
• affine invariant polynomials