  Instabilities resulting in Bose-Einstein condensation and/or modulation of
"displacements" in a system of quantum particles described by a two-state
Bose-Hubbard model (with an allowance for the interaction between particle
displacements on different lattice sites) are investigated. A possibility of
modulation, which doubles the lattice constant, as well as the uniform
displacement of particles from equilibrium positions are studied. Conditions
for realization of the mentioned instabilities and phase transitions into the
SF phase and into the "ordered" phase with frozen displacements are analyzed.
The behaviour of order parameters is investigated and phase diagrams of the
system are calculated both analytically (ground state) and numerically (at
non-zero temperatures). It is revealed that the SF phase can appear as an
intermediate one between the normal and "ordered" phases, while a supersolid
phase is thermodynamically unstable and does not appear. The relation of the
obtained results to the lattices with the double-well local potentials is
discussed.
