  The axion modified electrodynamics is usually used as a model for description
of possible violation of Lorentz invariance in field theory. The low-energy
manifestation of Lorentz violation can hopefully be observed in experiments
with the electromagnetic waves. It justifies the importance of study how a
small axion addition can modify the wave propagation. Although a constant axion
does not contribute to the dispersion relation at all, even a slowly varying
axion field destroys the light cone structure. In this paper, we study the wave
propagation in the axion modified electrodynamics in the framework of premetric
approach. In an addition to the modified dispersion relation, we derive the
axion generalization of the photon propagator in Feynman and Landau gauge. Our
consideration is free from the usual restriction of a usual restriction to the
constant gradient axion field. It is remarkable that the axion modified
propagator is hermitian. Consequently the dissipation effects absent even in
the phenomenological model considered here.
