The integrability of the 2-Toda lattice on a simple Lie algebra

We define the 2-Toda lattice on every simple Lie algebra gg, and we show its Liouville integrability. We show that this lattice is given by a pair of Hamiltonian vector fields, associated with a Poisson bracket which results from an RR-matrix of the underlying Lie algebra. We construct a big family of constants of motion which we use to prove the Liouville integrability of the system. We achieve the proof of their integrability by using several results on simple Lie algebras, RR-matrices, invariant functions and root systems.