The Adler Kostant Symes scheme in physics

The purpose of this material is to review the Adler Kostant Symes scheme as a theory which can be developped succesfully in different contexts. It was useful to describe some mechanical systems, the so called generalized Toda, and now it was proved to be a tool for the study of the linear approach to the motion of n uncoupled harmonic oscillators. The complete integrability of these systems has an algebraic description. In the original theory this is related to ad-invariant functions, but new examples show that new conditions should be investigated.