Radiative corrections in Lorentz violating extensions of QED have received considerable attention in the last years. In this talk I will address the case of fermions coupled to a constant background axial-vector and analise both classical and quantum aspects which prove to be relevant for the consistence of the theory. The modification of the mass-shell conditions in the free theory allows to determine bounds on the fermions' momenta which in turns determines the domain of integration for radiative corrections. We consider this to render the theory free of ambiguities, rather than formal arguments which may lack foundation in this context. Also, the appearance of an anomalous loss of gauge invariance and the role of finite temperature are discussed.