  In this article we discuss a weaker version of Liouville's theorem on the
integrability of Hamiltonian systems. We show that in the case of Tonelli
Hamiltonians the involution hypothesis on the integrals of motion can be
completely dropped and still interesting information on the dynamics of the
system can be deduced. Moreover, we prove that on the n-dimensional torus this
weaker condition implies classical integrability in the sense of Liouville. The
main idea of the proof consists in relating the existence of independent
integrals of motion of a Tonelli Hamiltonian to the size of its Mather and
Aubry sets. As a byproduct we point out the existence of non-trivial common
invariant sets for all Hamiltonians that Poisson-commute with a Tonelli one.
