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 Jacksons 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.
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