  The dynamics of the development of instability of the free surface of liquid
helium, which is charged by electrons localized above it, is studied. It is
shown that, if the charge completely screens the electric field above the
surface and its magnitude is much larger then the instability threshold, the
asymptotic behavior of the system can be described by the well-known 3D
Laplacian growth equations. The integrability of these equations in 2D geometry
makes it possible to described the evolution of the surface up to the formation
of singularities, viz., cuspidal point at which the electric field strength,
the velocity of the liquid, and the curvature of its surface assume infinitely
large values. The exact solutions obtained for the problem of the
electrocapillary wave profile at the boundary of liquid helium indicate the
tendency to a charge in the surface topology as a result of formation of
charged bubbles.
