  To investigate the phenomenon of Bose-Einstein condensation in perfect
crystals a hierarchy of equations for reduced density matrices that describes a
thermodynamically equilibrium quantum system is employed, the hierarchy being
obtained earlier by the author. The thermodynamics of a crystal with a
condensate and the one of a crystal with no condensate are constructed in
parallel, which is required for studying the phase transition involving
Bose-Einstein condensation. The transition is analysed also with the help of
the Landau theory of phase transitions which shows that a superfluid state can
result either from two consecutive phase transitions or from only one. To
demonstrate how the general equations obtained can be applied for a concrete
crystal the bifurcation method for solving the equations is utilized. New
results concerning properties of the condensate crystals at zero temperature
are obtained as well. In the concluding section, the physical concept of the
condensate is discussed.
