  We consider the following Hamiltonian equation on a special manifold of
rational functions, \[i\p\_tu=\Pi(|u|^2u)+\al (u|1),\ \al\in\R,\] where $\Pi $
denotes the Szeg\H{o} projector on the Hardy space of the circle $\SS^1$. The
equation with $\al=0$ was first introduced by G{\'e}rard and Grellier in
\cite{GG1} as a toy model for totally non dispersive evolution equations. We
establish the following properties for this equation. For $\al\textless{}0$,
any compact subset of initial data leads to a relatively compact subset of
trajectories. For $\al\textgreater{}0$, there exist trajectories on which high
Sobolev norms exponentially grow with time.
