  The purpose of this paper is to go further into the study of the quadratic
Szeg{\"o} equation, which is the following Hamiltonian PDE : $i \partial\_t u =
2J\Pi(|u|^2)+\bar{J}u^2$, $u(0, \cdot)=u\_0$, where $\Pi$ is the Szeg{\"o}
projector onto nonnegative modes, and $J = J(u)$ is the complex number given by
$J=\int\_\mathbb{T}|u|^2u$. We exhibit an infinite set of new conservation laws
$\{\ell\_k \}$ which are in involution. These laws give us a better
understanding of the "turbulent" behavior of certain rational solutions of the
equation : we show that if the orbit of a rational solution is unbounded in
some $H^s$, $s > 1/2$, then one of the $\ell\_k$'s must be zero. As a
consequence, we characterize growing solutions which can be written as the sum
of two solitons.
