  A mechanism of both formation of peaks in the density of states near the
Fermi surface and phase instabilities of nearly ideal degenerate Fermi gas in
low-dimensional optical lattices is proposed. According to this mechanism, peak
formation is caused by the quasi-classical quantization of the one- and
two-dimensional fermionic spectrum in the neighborhood of its extremal points
under interaction with an long-wave periodical perturbation. The new spectra
result in the instabilities with respect to spontaneous formation of an
equilibrium superstructure. In the one-dimensional case this happens for low
enough numbers of fermionic atoms. As a result of such transition, fermions
become localized (a transition of the metal-insulator type). In the
two-dimensional system the transition is possible for a nearly half-filled
band. In this case fermions are localized in the wave direction only. It is
briefly discussed the possible influence of the results obtained in the paper
on the superfluid transition temperature in high anisotropic lattices
possessing quasi-(one,two)-dimensional subsystems of fermionic atoms.
