  We show how a mass mixing matrix can be generated dynamically, for two
massless fermion flavours coupled to a Lorentz invariance violating (LIV) gauge
field. The LIV features play the role of a regulator for the gap equations, and
the non-analytic dependence of the dynamical masses, as functions of the gauge
coupling, allows to consider the limit where the LIV gauge field eventually
decouples from the fermions. Lorentz invariance is then recovered, to describe
the oscillation between two free fermion flavours, and we check that the finite
dynamical masses are the only effects of the original LIV theory. We also
discuss briefly a connection of our results with the case of Majorana neutrinos
in both, the standard model, where only left-handed (active) neutrinos are
considered, and extensions thereof, with sterile right-handed neutrinos.
