  A new variational method for studying the equilibrium states of an
interacting particles system has been proposed. The statistical description of
the system is realized by means of a density matrix. This method is used for
description of conduction electrons in metals. An integral equation for the
electron distribution function over wave vectors has been obtained. The
solutions of this equation have been found for those cases where the
single-particle Hamiltonian and the electron interaction Hamiltonian can be
approximated by a quite simple expression. It is shown that the distribution
function at temperatures below the critical value possesses previously unknown
features which allow to explain the superconductivity of metals and presence of
a gap in the energy spectrum of superconducting electrons.
