On the Lie-Poisson structure of the nonlinearized discrete eigenvalue problem

A 3x3 discrete eigenvalue problem and corresponding discrete soliton equati ons are proposed. Under a constraint between the potentials and eigenfuncti ons, the 3x3 discrete eigenvalue problem is nonlinearized into an integrabl e Poisson map with a Lie-Poisson structure. Further, a reduction of the Lie -Poisson structure on the co-adjoint orbit yields the standard symplectic s tructure. The Poisson map is reduced to the usual symplectic map when it is restricted on the leaves of the symplectic foliation. (C) 2000 American In stitute of Physics. [S0022-2488(00)00408-4].