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Firstly, a hierarchy of integrable lattice equations and its bi-Hamiltonian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.

DECOMPOSING A NEW NONLINEAR DIFFERENTIAL-DIFFERENCE SYSTEM UNDER A BARGMANN IMPLICIT SYMMETRY CONSTRAINT