  The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems
for quadratic Hamiltonians of $\mathbb R^{2n}$ on coadjoint orbits of the
Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie
algebra $\mathfrak g$ that admits an ad-invariant metric. Its quadratic induces
the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that
one on $\mathbb R^{2n}$. 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 $\mathfrak g$
is related to the commutativity of a family of derivations of the
2n+1-dimensional Heisenberg Lie algebra $\mathfrak h_n$. Therefore the complete
integrability is related to the existence of an n-dimensional abelian
subalgebra of certain derivations in $\mathfrak h_n$. For instance, the motion
of n-uncoupled harmonic oscillators near an equilibrium position can be
described with this setting.
