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

A 3 x 3 discrete eigenvalue problem associated with Toda hierarchy is prese nted. After the nonlinearization procedure, the 3 x 3 discrete eigenvalue p roblem is turned into an integrable Poisson map on the Poisson manifold R-3 N With a Lie-Poisson structure. As a reduction of the Lie-Poisson structure on the co-adjoint orbit, the standard symplectic structure on the symplect ic manifold R-2N is obtained. The Poisson map restricted on the leaves of t he symplectic foliation is reduced to a usual symplectic map, which is exac tly the nonlinearized 2 x 2 eigenvalue problem. (C) 2000 Elsevier Science B .V. All rights reserved.