A mean field approach to $Z_N$-enhanced generalized May-Leonard models

May-Leonard (ML) models have been used to describe the rich dynamics of a range of systems in biology and ecology. In this report we study a class of extended cyclic ML models of N species in the mean field limit, enhanced with $Z_N$ symmetry, and investigate the space of their (unstable) coexistence fixed points. We start with a brief review of the well studied ML model of three species, expand on the generalized class and provide expressions for the unstable invariant manifold near single fixed points of a subclass of the mentioned extensions. For the purely cyclic ML model with an odd number of species we derive the complex Ginzburg-Landau normal form.