  The stability of the excitonic condensation at low temperature driven by a
coupling of electrons to vibrational degrees of freedom in semimetal
two-dimensional electronic system is discussed. In the framework of the
unrestricted Hartree-Fock approximation, we derive a set of equations to
determine both the excitonic condensate order parameter and lattice
displacement self-consistently. By lowering temperature we find out a
semimetal-insulator transition in the system if the coupling is large enough.
The insulating state typifies an excitonic condensation accompanied by a finite
lattice distortion. Increasing temperature, both excitonic condensate order
parameter and the lattice distortion decrease and then disappear in the same
manner. Microscopic analysis in momentum space strongly specifies that the
excitonic condensate driven by the lattice distortion favours the BCS type.
