  We describe a Horava-Lifshitz-like reformulated four-fermion Gross-Neveu
model describing the dynamics of two-component spinors in (2+1)-dimensional
space-time. Within our study, we introduce the Lagrange multiplier, study the
gap equation (including the finite temperature case) which turns out to display
essentially distinct behaviors for even and odd values of the critical exponent
z, and show that the dynamical parity breaking occurs only for the odd z. We
demonstrate that for any odd z, there exists a critical temperature at which
the dynamical parity breaking disappears. Besides of this, we obtain the
effective propagator and show that the resulting effective theory is
renormalizable within the framework of the 1/N expansion for all values of z.
As one more application of the dynamical parity breaking, we consider coupling
of the vector field to the fermions in the case of a simplified spinor-vector
coupling and discuss the generation the Chern-Simons term.
