Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles

Higher-rank solutions of the equations of the two-dimensionalized Toda lattice are constructed. The construction of these solutions is based on the theory of commuting difference operators, which is developed in the first part of the paper. It is shown that the problem of recovering the coefficients of commuting operators can be effectively solved by means of the equations of the discrete dynamics of the Tyurin parameters characterizing the stable holomorphic vector bundles over an algebraic curve