  We consider the following Hamiltonian equation on the $L^2$ Hardy space on
the circle, $$i\partial_tu=\Pi(|u|^2u) ,$$ where $\Pi $ is the Szeg\"o
projector. This equation can be seen as a toy model for totally non dispersive
evolution equations. We display a Lax pair structure for this equation. We
prove that it admits an infinite sequence of conservation laws in involution,
and that it can be approximated by a sequence of finite dimensional completely
integrable Hamiltonian systems. We establish several instability phenomena
illustrating the degeneracy of this completely integrable structure. We also
classify the traveling waves for this system.
