Affine invariant conditions for a class of differential polynomial cubic systems

Abstract

In this article the affine invariant criteria constructed in terms of algebraic polynomials with coefficients $\tilde a \in \mathcal R^{20}$ for a class of cubic systems are established. We are focused on non-degenerate real cubic systems with 7 invariant straight lines, considering the line at infinity and their multiplicities and possesing four real singularities at infinity. Additionally, the only configurations of the type $(3, 3)$ of mentioned systems are considered and we denote this class by $CSL_{(3,3)}^{4r\infty}$. In \cite{Buj-Sch-Vul-EJDE-2021} the existence of exactly 14 configurations of invariant straight lines for systems in $CSL_{(3,3)}^{4r\infty}$ was proved. Here we complete this classification by determining necessary and sufficient conditions for the realization of each one of the 14 configurations in terms of affine invariant polynomials.