  The Hohenberg-Martin dilemma of conserving versus gapless theories for
systems with Bose-Einstein condensate is considered. This dilemma states that,
generally, a theory characterizing a system with broken global gauge symmetry,
which is necessary for Bose-Einstein condensation, is either conserving, but
has a gap in its spectrum, or is gapless, but does not obey conservation laws.
In other words, such a system either displays a gapless spectrum, which is
necessary for condensate existence, but is not conserving, which implies that
it corresponds to an unstable system, or it respects conservation laws,
describing a stable system, but the spectrum acquires a gap, which means that
the condensate cannot appear. An approach is described, resolving this dilemma,
and it is shown to give good quantitative agreement with experimental data.
Calculations are accomplished in the Hartree-Fock-Bogolubov approximation.
