Classification of traveling waves for a quadratic Szeg{\"o} equation

We give a complete classification of the traveling waves of the following quadratic Szeg{"o} equation : $i \partial_t u = 2J\Pi(|u|^2)+\bar{J}u^2, \quad u(0, \cdot)=u_0$, and we show that they are given by two families of rational functions, one of which is generated by a stable ground state. We prove that the other branch is orbitally unstable.