We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle,$$i{\partial }_{t}u={\Pi \left(\right|u|}^{2}u),$$where $\Pi$ is the Szegő 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...