We consider the excitonic insulator state (often associated with electronic ferroelectricity), which arises on the phase diagram of an extended spinless Falicov--Kimball model (FKM) at half-filling. Within the Hartree--Fock approach, we calculate the spectrum of  low-energy collective excitations in this state up to second order in the narrow-band hopping and/or hybridisation. This allows to probe the mean-field stability of the excitonic insulator. The latter is found to be unstable when the case of the pure FKM (no hybridisation with a fully localised band) is approached. The instability is due to the presence of another, lower-lying ground state and {\it not} to the degeneracy of the excitonic phase in the pure FKM. The excitonic phase, however, may be stabilised further away from the pure FKM limit. In this case, the low-energy excitation spectrum  contains new information about  the properties of the excitonic condensate (including the strongly suppressed critical temperature).

[1] D. I. Golosov, Phys. Rev. B vol. 86, 155134 (2012)