Finite reduction groups, automorphic Lie algebras and applications to integrable systems

We discuss structure and some classification results for automorphic Lie algebras associated with finite reduction groups. We construct Lax representations (zero curvature representations) corresponding to these automorphic algebras which lead to systems of integrable partial differential equations, as well as we construct a number of reduction group invariant Darboux transformations which lead to integrable differential difference and finite difference systems.