  We prove two statements about the long time dynamics of integrable
Hamiltonian systems. In classical mechanics, we prove the microcanonical
version of the Generalized Gibbs Ensemble (GGE) by mapping it to a known
theorem and then extend it to the limit of infinite number of degrees of
freedom. In quantum mechanics, we prove GGE for maximal Hamiltonians - a class
of models stemming from a rigorous notion of quantum integrability understood
as the existence of conserved charges with prescribed dependence on a system
parameter, e.g. Hubbard U, anisotropy in the XXZ model etc. In analogy with
classical integrability, the defining property of these models is that they
have the maximum number of independent integrals. We contrast their dynamics
induced by quenching the parameter to that of random matrix Hamiltonians.
