  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.
