  In this paper, we discuss various possible schemes for the perturbative
generation of the axion-photon interaction term in different Lorentz-breaking
extensions of QED, involving operators with mass dimensions up to 5. We
demonstrate explicitly that there are only a few schemes allowing to generate a
finite axion-photon interaction term from one-loop radiative corrections, and
within all these schemes, the generated term turns out to be ambiguous.
