  In this paper, we prove a version of Arnol'd-Liouville theorem for C 1
commuting Hamiltonians. We show that the Lipschitz regularity of the foliation
by invariant Lagrangian tori is crucial to determine the Dynamics on each
Lagrangian torus and that the C 1 regularity of the foliation by invariant
Lagrangian tori is crucial to prove the continuity of Arnol'd-Liouville
coordinates. We also explore various notions of C 0 and Lipschitz
integrability.
