The Lie-Poisson representation of the nonlinearized eigenvalue problem of the Kac-van Moerbeke hierarchy

The Kac-van Moerbeke hierarchy is studied by a 3 x 3 discrete eigenvalue pr oblem and the corresponding nonlinearized one an integrable Poisson map wit h a Lie-Poisson structure is also presented. Moreover, the 2 x 2 nonlineari zed eigenvalue problem associated with the Kac-van Moerbeke hierarchy is pr oved to be a reduction of the Poisson map on the leaves of the symplectic f oliation. (C) 2001 Elsevier Science B.V. All rights reserved.