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 $gl(n,{\mathbb R})$. This system was obtained by reducing the space of maps from $Z_n$ to $GL(n,{\mathbb R})$ endowed with a structure of a pair of Lie-algebroids.