In this paper, we prove a version of Arnol$ '$d-Liouville theorem for $ C^2$ Hamiltonians having enough $ 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.