A Hierarchy of Lax Integrable Lattice Equations, Liouville Integrability and a New Integrable Symplectic Map**The project supported by the Scientific Research Award Foundation for Outstanding Young and Middle-Aged Scientists of Shandong Province of China

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A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.