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We consider the structure of halos of the axionic dark matter, which surroundmassive relativistic objects with static spherically symmetric gravitationalfield and monopole-type magneto-electric fields. We work with the model ofpseudoscalar field with the extended periodic potential, which depends onadditional arguments proportional to the moduli of the Killing vectors; in ourapproach they play the roles of model guiding functions. The covariant model ofthe axion field with this modified potential is equipped with the extendedformalism of the Killing vector fields, which is established in analogy withthe formalism of the Einstein-aether theory, based on the introduction of aunit timelike dynamic vector field. We study the equilibrium state of the axionfield, for which the extended potential and its derivative vanish, andillustrate the established formalism by the analysis of two-level axionic darkmatter profiles, for which the stage delimiters relate to the critical valuesof the modulus of the timelike Killing vector field.

New application of the Killing vector field formalism: Modified periodic potential and two-level profiles of the axionic dark matter distribution