  We study instabilities occurring in the electron system whose Fermi surface
has flat regions on its opposite sides. Such a Fermi surface resembles Fermi
surfaces of some high-$T_c$ superconductors. In the framework of the parquet
approximation, we classify possible instabilities and derive
renormalization-group equations that determine the evolution of corresponding
susceptibilities with decreasing temperature. Numerical solutions of the
parquet equations are found to be in qualitative agreement with a ladder
approximation. For the repulsive Hubbard interaction, the antiferromagnetic
(spin-density-wave) instability dominates, but when the Fermi surface is not
perfectly flat, the $d$-wave superconducting instability takes over.
