  We consider the structure of halos of the axionic dark matter, which surround
massive relativistic objects with static spherically symmetric gravitational
field and monopole-type magneto-electric fields. We work with the model of
pseudoscalar field with the extended periodic potential, which depends on
additional arguments proportional to the moduli of the Killing vectors; in our
approach they play the roles of model guiding functions. The covariant model of
the axion field with this modified potential is equipped with the extended
formalism of the Killing vector fields, which is established in analogy with
the formalism of the Einstein-aether theory, based on the introduction of a
unit timelike dynamic vector field. We study the equilibrium state of the axion
field, for which the extended potential and its derivative vanish, and
illustrate the established formalism by the analysis of two-level axionic dark
matter profiles, for which the stage delimiters relate to the critical values
of the modulus of the timelike Killing vector field.
