Integrable dynamics of Toda type on square and triangular lattices

In a recent paper we constructed an integrable generalization of the Toda law on the square lattice. We construct other examples of integrable dynamics of Toda type on the square lattice, as well as on the triangular lattice, as nonlinear symmetries of the discrete Laplace equations on square and triangular lattices. We also construct the tau-function formulations and Darboux-Backlund transformations of these dynamics.