This is the second talk devoted to the construction of global Birkhoff variables for the Benjamin-Ono evolution on the torus, in collaboration with Thomas Kappeler. We establish a Lax pair identity for the Hamiltonian evolution associated to the generating function defining the Benjamin-Ono hierarchy. Using this structure, we prove canonical Poisson brackets identities for the Birkhoff coordinates, and finally that the Birkhoff map is bijective. A special role is played by the class of finite gap potentials, which we study in detail.