The binary nonlinearization of generalized Toda hierarchy by a special choice of parameters

A new discrete matrix spectral problem with two arbitrary constants is introduced. The corresponding 2-parameter hierarchy of integrable lattice equations, which can be reduced to the hierarchy of Toda lattice, is obtained by discrete zero curvature representation. Moreover, the Hamiltonian structure and a hereditary operators are deduced by applying the discrete trace identity. Finally, an integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonlinearization for the resulting hierarchy by a special choice of parameters.

Research highlights
► A new discrete matrix spectral problem with two arbitrary constants is introduced.
► The corresponding 2-parameter integrable lattice hierarchy is obtained.
► The problem on Bargmann symmetry constraint is studied extensively and instructively.
► Significant improvements was observed compared with the reported results.