An integrable normal form for water waves in in nite

We consider the Birkhoo normal form for the water wave problem posed in a uid of innnite depth, with the starting point of our analysis a version of the Hamiltonian given by V.E. Zakharov. We verify that in the fourth order normal form, the coeecients vanish for all non-generic resonant terms, and we show that the resulting truncated system is completely integrable. In contrast we show that there are resonant fth order terms with nonvanishing coeecients, answering to the negative the conjecture of A.I. Dyachenko & Zakharov on the integrability of free surface hydrodynamics. We analyse all solutions of the fourth order integrable system. In particular, the bifurcation structures of standing wave and travelling wave solutions are remarkably explicit due to the normal form.