  The bilayer systems exhibit the Bose-Einstein condensation of excitons that
emerge due to Coulomb pairing of electrons belonging to one layer with the
holes belonging to the other layer. Here we present the microscopic derivation
of the dynamic equation for the condensate wave function at a low density of
electron-hole ($e-h$) pairs in a strong magnetic field perpendicular to the
layers and an electric field directed along the layers. From this equation we
obtain the dispersion law for collective excitations of the condensate and
calculate the electric charge of the vortex in the exciton condensate. The
critical interlayer spacing, the excess of which leads to a collapse of the
superfluid state, is estimated. In bilayer systems with curved conducting
layers, the effective mass of the $e-h$ pair becomes the function of the $e-h$
pair coordinates, the regions arise, where the energy of the $e-h$ pair is
lowered (exciton traps), and lastly $e-h$ pairs can gain the polarization in
the basal plane. This polarization leads to the appearance of quantized
vortices even at zero temperature.
