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.
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