A Gentle (without Chopping) Approach to the Full Kostant-Toda Lattice
  In this paper we propose a new algorithm for obtaining the rational integrals of the full Kostant-Toda lattice. This new approach is based on a reduction of a bi-Hamiltonian system on gl(n,R). This system was obtained by reducing the space of maps from Z_n to GL(n,R) endowed with a structure of a pair of Lie-algebroids. 
Pantelis A. Damianou and Franco Magri