  The group classification of models of axion electrodynamics with arbitrary
self interaction of axionic field is carried out. It is shown that extensions
of the basic Poincar\'e invariance of these models appear only for constant and
exponential interactions. The related conservation laws are discussed. Using
the In\"on\"u-Wigner contraction the non-relativistic limit of equations of
axion electrodynamics is found. An extended class of exact solutions for the
electromagnetic and axion fields is obtained. Among them there are solutions
including up to six arbitrary functions. In particular, solutions which
describe propagation with velocities faster than the velocity of light are
found. These solutions are smooth and bounded functions which correspond to
positive definite and bounded energy density.
