  A theory of equilibrium states of electrons above a liquid helium surface in
the presence of an external clamping field is built based on the first
principles of quantum statistics for the system of many identical
Fermi-particles. The approach is based on the variation principle modified for
the considered system and on Thomas-Fermi model. In terms of the developed
theory we obtain the self-consistency equations that connect the parameters of
the system description, i.e., the potential of a static electric field, the
distribution function of electrons and the surface profile of a liquid
dielectric. The equations are used to study the phase transition of the system
to a spatially periodic state. To demonstrate the capabilities of the proposed
method, the characteristics of the phase transition of the system to a
spatially periodic state of a trough type are analyzed.
