  The paper is devoted to the proof of the uniqueness theorem for solution of
the equation for the non-local ionization source in a glow discharge and a
hollow cathode in general 3D geometry. The theorem is applied to wide class of
electric field configurations, and to the walls of discharge volume, which have
a property of incomplete absorption of the electrons. Cathode is regarded as
interior singular source, which is placed arbitrarily close to the wall. The
existence of solution is considered also. During the proof of the theorem many
of useful structure formulae are obtained. Elements of the proof structure,
which have arisen, are found to have physical sense. It makes clear physical
construction of non-local electron avalanche, which builds a source of
ionization in glow discharge at low pressures. Last has decisive significance
to understand the hollow cathode discharge configuration and the hollow cathode
effect.
