Are Equations of Free-Surface Hydrodynamics on Deep Water Integrable?

We show that the Euler equations describing the unsteady potential flow of a two-dimensional deep fluid with a free surface in the absence of gravity and surface tension can be integrated exactly under a special self-similar choice of boundary conditions at infinity. Problem of exact integrability of liquid with natural boundary conditions remains open although there exist analytical and numerical arguments supporting this conjecture, in particular, existence of indefinite number (depending on the initial conditions) of commuting integrals of motion. Another strong argument is the exact cancellation of coefficients of non-trivial four wave interactions. This cancellation explains strong slowdown of stochastisation process observed in numerical experiments in the 70-ies of the previous century. Modern experiments show long existence of breather which decays in short time for non-integrable case. Proof of integrability of deep water equations would be a discovery of a completely new class of integrable systems.