  This paper studies the existence of invariant smooth Lagrangian graphs for
Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli
Hamiltonians with n independent but not necessarily involutive constants of
motion and obtain two theorems reminiscent of the Liouville-Arnold theorem.
Moreover, we also obtain results on the structure of the configuration spaces
of such systems that are reminiscent of results on the configuration space of
completely integrable Tonelli Hamiltonians.
