Fibration of the phase space for the Korteweg-de Vries equation

In this article we prove that the fibration of $L^ 2(S^ 1)$ by potentials which are isospectral for the 1-dimensional periodic Schrödinger equation, is trivial. This result can be applied, in particular, to $N$-gap solutions of the Korteweg-de Vries equation (KdV) on the circle: one shows that KdV, a completely integrable Hamiltonian system, has global action-angle variables