The integrability of the periodic Full Kostant-Toda on a simple Lie   algebra
  We define the periodic Full Kostant-Toda on every simple Lie algebra, and show its Liouville integrability. More precisely we show that this lattice is given by a Hamiltonian vector field, associated to a Poisson bracket which results from an R-matrix. We construct a large family of constant of motion which we use to prove the Liouville integrability of the system with the help of several results on simple Lie algebras, R-matrix, invariant functions and root systems. 
Khaoula Ben Abdeljelil
math.AG