  In the framework of the Einstein-Maxwell-axion-aether theory we establish the
model, the Lagrangian of which contains the sin-type generalization of the term
describing the axion-photon coupling, and the axionically induced cosine-type
modification of the term attributed to the dilaton-photon interactions. The
extension of the axion-dilaton-aether electrodynamics is inspired by the
Jackson's idea concerning the internal symmetry of the equations of
electromagnetism. The application of the extended theory to the anisotropic
homogeneous cosmological model of the Bianchi-I type is considered. The exact
solutions to the model evolutionary equations are obtained for the case, when
the axionic dark matter is in the state of equilibrium, which is characterized
by vanishing potential of the pseudoscalar field and its first derivative. The
state of the axion-photon system, which is of a new type and is indicated as a
dynamic equilibrium, is studied in the framework of electrodynamics with
axionic non-linearity. We show that the nonlinear axion-photon interactions can
mimic the dilaton-photon coupling. We discuss the stability of the model with
respect to homogeneous fluctuations of the axion field.
