Integrable dynamics of Toda-type on the square and triangular lattices

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In a recent paper we constructed an integrable generalization of the Toda law on the square lattice. In this paper 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 the square and triangular lattices. We also construct the τ - function formulations and the Darboux-Bäcklund transformations of these novel dynamics.