The family of cubic differential systems with two real and two complex distinct infinite singularities and invariant straight lines of the type $(2,2,2)$

Cristina Bujac ; Nicolae Vulpe

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2024), pp. 84-99.

Résumé

We denote by ${\mathbf{CSL}}_7$ the family of cubic differential systems possessing invariant straight lines, finite and infinite, of total multiplicity exactly seven. In a sequence of papers the study of the subfamily of cubic systems belonging to ${\mathbf{CSL}}_7$ with $4$ real distinct singular points at infinity was reached. The goal of this article is to continue the study of the geometric configurations of invariant lines of ${\mathbf{CSL}}_7$ with two real and two complex distinct infinite singularities and invariant lines in the configuration of the type $(2,2,2)$. We proved that there exists only one configuration of invariant straight lines belonging to the class mentioned above. In addition, we construct invariant affine criteria for the realization of the obtained configuration.

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