Family of quadratic differential systems with two invariant conics: parabola and hyperbola

Consider the class QS of all non-degenerate planar quadratic differential systems and its subclass QSPH of all its systems possessing an invariant parabola and an invariant hyperbola simultaneously. We investigate all possible configurations of these two invariant conics and invariant straight lines that systems in QSPH could possess and their geometric properties encoded in such configurations. The classification presented here is taken modulo the action of the group of real affine transformations and time rescaling, and it is given in terms of affine invariant polynomials. It yields a total of 43 distinct configurations. The obtained classification is an algorithm that makes it possible for any given real quadratic differential system in QSPH to specify its configuration of invariant parabolas and hyperbolas and straight lines. This work will prove helpful in studying the integrability of the systems in QSPH .