  We study renormalization group flows in the Lifshitz-like $N$-flavour four
fermi model discussed in 0905.2928. In the large-$N$ limit, a nontrivial flow
occurs in only one of all possible marginal couplings and one relevant
coupling, which provides the scale for Lorentz invariance violations. We
discuss in detail the phase diagram and RG flows in the space of couplings,
which includes the Lifshitz fixed point, the free field fixed point and a new
fixed point characterized by $z=1$ scaling and a violation of Lorentz
invariance, which cannot be tuned away by adjusting a parameter. In the broken
symmetry phase, the model flows from the $z=3$ Lifshitz-like fixed point in the
ultraviolet to this new fixed point in the infrared. However, in a modified
version of the present model, which has an effective ultraviolet cut-off much
smaller than the Lorentz invariance violating scale, the infrared behaviour is
governed by an approximately Lorentz invariant theory, similar to the low
energy limit of the usual relativistic Nambu$-$Jona-Lasinio model. Such a
modified model could be realized by a supersymmetric version of the present
model.
