Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations

We give a family of planar polynomial differential systems whose limit cycles can be explicitly described using polar coordinates. Moreover, we characterize the multiplicity of each one of the limit cycles whenever they exist. The given family of planar polynomial differential systems can have at most two limit cycles, counted with multiplicity. As an application of this result we give an example of a planar polynomial differential system with two explicit limit cycles, one of them algebraic and the other one non-algebraic.