The nonlinearized eigenvalue problem of the Toda hierarchy in the Lie–Poisson framework

A 3×3 discrete eigenvalue problem associated with Toda hierarchy is presented. After the nonlinearization procedure, the 3×3 discrete eigenvalue problem is turned into an integrable Poisson map on the Poisson manifold R3N with a Lie–Poisson structure. As a reduction of the Lie–Poisson structure on the co-adjoint orbit, the standard symplectic structure on the symplectic manifold R2N is obtained. The Poisson map restricted on the leaves of the symplectic foliation is reduced to a usual symplectic map, which is exactly the nonlinearized 2×2 eigenvalue problem.