A discrete Garnier type system from symmetry reduction on the lattice

A symmetry reduction of the lattice modified Boussinesq system is studied. The full group of Lie point symmetries of the relevant system is retrieved and certain group invariant solutions are considered by using an accessional generalized symmetry. It is demonstrated that the symmetry reduction leads to a coupled set of second-order nonlinear non-autonomous ordinary difference equations involving six free parameters, generalizing to higher order some of the known discrete analogues of the Painlevé VI equation. The corresponding isomonodromic deformation problem is constructed through the symmetry reduction as well.