  Using coherent-state formalism (the Keldysh formalism), the article describes
a transition from a homogeneous superfluid state to a supersolid state in a
two-dimensional dilute gas of electron-hole pairs with spatially separated
components. Such a transition is heralded by the appearance of a roton-type
minimum in the collective excitation spectrum, which touches the abscissa axis
as the distance between the layers or the pair density increases. This signals
the instability of the system with respect to the appearance of a spatial
modulation of the pair density. It has been found that a first-order transition
to a hexagonal supersolid phase takes place a little earlier. A theory without
phenomenological constants has been developed for an arbitrary relation between
the effective masses of an electron and a hole. A phase diagram for the system
has been plotted in the variables "the chemical potential of pairs - the
distance between the layers". It has been shown that there is a jump in the
average density of the condensate during the phase transition. It has been
established that with an increase in the chemical potential, the inhomogeneous
phase breaks up into high-density regions surrounded by lines at which the
density becomes zero, with these lines forming a continuous network.
