Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x² + y² + 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincaré disc.