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The bilayer systems exhibit the Bose-Einstein condensation of excitons thatemerge due to Coulomb pairing of electrons belonging to one layer with theholes belonging to the other layer. Here we present the microscopic derivationof the dynamic equation for the condensate wave function at a low density ofelectron-hole ($e-h$) pairs in a strong magnetic field perpendicular to thelayers and an electric field directed along the layers. From this equation weobtain the dispersion law for collective excitations of the condensate andcalculate the electric charge of the vortex in the exciton condensate. Thecritical interlayer spacing, the excess of which leads to a collapse of thesuperfluid state, is estimated. In bilayer systems with curved conductinglayers, 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 islowered (exciton traps), and lastly $e-h$ pairs can gain the polarization inthe basal plane. This polarization leads to the appearance of quantizedvortices even at zero temperature.Comment: 8 page

Dynamic equation for quantum Hall bilayers with spontaneous interlayer coherence: The low-density limit