  We continue the study of the following Hamiltonian equation on the Hardy
space of the circle, $$i\partial _tu=\Pi(|u|^2u)\ ,$$ where $\Pi $ denotes the
Szeg\"o projector. This equation can be seen as a toy model for totally non
dispersive evolution equations. In a previous work, we proved that this
equation admits a Lax pair, and that it is completely integrable. In this
paper, we construct the action-angle variables, which reduces the explicit
resolution of the equation to a diagonalisation problem. As a consequence, we
solve an inverse spectral problem for Hankel operators. Moreover, we establish
the stability of the corresponding invariant tori. Furthermore, from the
explicit formulae, we deduce the classification of orbitally stable and
unstable traveling waves.
