On a Trivial Family of Noncommutative Integrable Systems

We discuss trivial deformations of the canonical Poisson brackets associated with the Toda lattices, relativistic Toda lattices, Henon–Heiles, rational Calogero–Moser and Ruijsenaars–Schneider systems and apply one of these deformations to construct a new trivial family of noncommutative integrable systems.