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The relation between the broken rotational symmetry of a system and the topology of its Fermi surface is studied for the two-dimensional system with the quasiparticle interaction f(p, p') having a sharp peak at |p − p'| = q0. It is shown that, in the case of attraction and q0 = 2pF the first instability manifesting itself with the growth of the interaction strength is the Pomeranchuk instability. This instability appearing in the L = 2 channel gives rise to a second order phase transition to a nematic phase. The Monte Carlo calculations demonstrate that this transition is followed by a sequence of the first and second order phase transitions corresponding to the changes in the symmetry and topology of the Fermi surface. In the case of repulsion and small values of q0, the first transition is a topological transition to a state with the spontaneously broken rotational symmetry, namely, corresponding to the nucleation of L ≃ π(pF/q0 − 1) small hole pockets at the distance pF − q0 from the center and the deformation of the outer Fermi surface with the characteristic multipole number equal to L. At q0 → 0, when the model under study transforms to the two-dimensional Nozieres model, the multipole number characterizing the spontaneous deformation is L → ∞, whereas the infinitely folded Fermi curve acquires the Hausdorff dimension D = 2 which corresponds to the state with the fermion condensate.

The model of dense neutron matter has been considered, where the topological rearrangement of the ground state of the system of Landau quasiparticles, which is associated with the appearance of the second sheet of the Fermi surface, occurs through two different scenarios. The rearrangement scenario depends on the relation between the wave vector q c of critical spin-isospin fluctuations and the Fermi momentum p F. Rearrangement at q c p F occurs with the stepwise appearance of a bubble with a radius of about 0.5p F in the filled Fermi sphere.

The role of an effective mass in the ab initio BCS equation for a pairing gap in atomic nuclei has been analyzed.