Explicit formulas for two limit cycles of a family of planar differential systems

In this work, we study a family of multi-parameter polynomial differential systems of degree eleven. We prove that the considered family has an invariant algebraic curve, which is given explicitly. Subsequently, we demonstrate the integrability of these systems and derive an explicit expression for a first integral. Moreover, we provide sufficient conditions for the systems to possess two explicitly given limit cycles. The applicability of our results is illustrated by a concrete example.