The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems
for quadratic Hamiltonians of R2n on coadjoint orbits of the
Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie
algebra g that admits an ad-invariant metric. Its quadratic induces
the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that
one on R2n. This system is a Lax pair equation whose solution can
be computed with help of the Adjoint representation. For a certain class of
functions, the Poisson commutativity on the coadjoint orbits in g
is related to the commutativity of a family of derivations of the
2n+1-dimensional Heisenberg Lie algebra hn​. Therefore the complete
integrability is related to the existence of an n-dimensional abelian
subalgebra of certain derivations in hn​. For instance, the motion
of n-uncoupled harmonic oscillators near an equilibrium position can be
described with this setting.Comment: 17 pages, it contains a theory about small oscillations in terms of
the AKS schem