Self-Dual Systems, their Symmetries and Reductions to the Bogoyavlensky Lattice

We recently introduced a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In particular, we introduced a subclass, which we called "self-dual". In this paper we discuss the continuous symmetries of these systems, their reductions and the relation of the latter to the Bogoyavlensky equation.