  We consider the following Hamiltonian equation on the $L^2$ Hardy space on
the circle $S^1$ , $$i\partial\_ t u = \Pi(|u|^ 2 u) + \alpha(u|1) , \alpha
\in\mathbb{R} ,$$ where $\Pi$ is the Szeg\H{o} projector. The above equation
with $\alpha= 0$ was introduced by G{\'e}rard and Grellier as an important
mathematical model [5, 7, 3]. In this paper, we continue our studies started in
[22], and prove our system is completely integrable in the Liouville sense. We
study the motion of the singular values of the related Hankel operators and
find a necessary condition of norm explosion. As a consequence, we prove that
the trajectories of the solutions will stay in a compact subset, while more
initial data will lead to norm explosion in the case $\alpha>0$.
