Abstract
By starting from a new discrete spectral problem, the Toda lattice is derived through the discrete zero curvature equation. Applying the discrete variational identity to the spectral problem will also reach to the bi-Hamiltonian structure of the Toda lattice. Under a higher-order Bargmann symmetry constraint, the new Lax pairs and the adjoint Lax pairs are nonlinearized into integrable symplectic maps and finite-dimensional Liouville integrable Hamiltonian systems. Finally, a Bäcklund transformation of the Toda lattice is obtained.
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