Global dynamics analysis of a class of quadratic polynomial differential systems with an invariant plane in R3

In this paper we study a class of quadratic differential systems in R3. More precisely, for three distinct families of parameter sets, such differential systems exhibit the property of having an invariant plane. The first family exhibits a first integral in the form of H(x, y) = ax + by ensures the invariance of every plane ax + by = constant. Consequently, we describe the phase portraits of the system on each of such planes in the Poincaré disc. The second family has a Darboux invariant of the form I(x, y, t) = d0 + d1x + d2ye-k0t which will allow us to describe the phase portraits on the Poincaré ball. Unlike the first two families, the third family lacks both a first integral and a Darboux invariant. Nevertheless, we present a detailed analysis of the phase portraits of these systems on the invariant plane using the Poincaré disc.