Is free surface hydrodynamics integrable?

I study the potental flow of ideal incompressible infinitely deep fluid with free surface in two dimensional space. Gravity and capilliarity might be included. The domain filled with fluid is conformally mapped onto the lower half-plane. The fluid dynamics is governed now by two evolutional intergo-differential equations having non-canonical Hamiltionian structure. How many motions contants are preserved by this dynamical system?
We show that besides the trivial constants (mass, momentum, energy) this system preserves indefinite number of extra motion constant. Their number depends on initial data. The function realizing the conformal mapping has moving cuts and moving zeros in the upper half plane. We show that each moving zero generates four independent extra motion constants. This fact leads to conjecture that the whole system is integrable, but this statement is not yet proven.