  By means of the Lee-Shrock transformation we generalize the 2D Gross-Neveu
(GN$_2$) model to a U(1) gauge theory with charged fermion and scalar fields in
2D ($\chi U \phi_2$ model). The $\chi U \phi_2$ model is equivalent to the
GN$_2$ model at infinite gauge coupling. We show that the dynamical fermion
mass generation and asymptotic freedom in the effective four-fermion coupling
persist also when the gauge coupling decreases. These phenomena are not
influenced by the XY$_2$ model phase transition at weak coupling. This suggests
that the $\chi U \phi_2$ model is in the same universality class as the GN$_2$
model and thus renormalizable.
