A new integrable symplectic map and the lie point symmetry associated with nonlinear lattice equations

A discrete matrix spectral problem is proposed, the hierarchy of discrete integrable system is inferred, which are Liouville integrable. And the Hamiltonian structures of the hierarchy are constructed. A family of finite-dimensional completely integrable systems and a new integrable symplectic map are provided in terms of the binary nonlinearity of spectral problem. In particular, two explicit formulations are acquired under the condition of the bargmann constraints. After that, the symmetry of the discrete integrable systems is given on the basis of the seed symmetry and its prolongation. Moreover, the solution of the discrete lattice equation can be gained by the way of the infinitesimal generator.