A hierarchy of discrete Hamiltonian equations and its binary nonlinearization by symmetry constraint

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A discrete matrix spectral problem is introduced, and a hierarchy of nonlinear lattice equations is derived. It is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. An integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonlinearization of the Lax pair and adjoint Lax pair for the resulting hierarchy. The binary Bargmann symmetry constraint leads to Bäcklund transformation for the resulting nonlinear integrable lattice equations.