Uniqueness problem of static axially symmetric black hole in magnetic Universe filled with { \it dark matter} component has been considered. The { \it dark matter} model comprises the additional $U(1)$-gauge field ({ \it dark photon}) interacting with Maxwell one by the kinetic mixing term. We show that all the solutions of Einstein-Maxwell { \it dark photon} gravity subject to the same boundary and regularity conditions authorize the only static axially symmetric black hole solutions with non-vanishing time and azimuthal components of Maxwell and { \it hidden} sector gauge fields, say Schwarzschild-like black hole immersed in { \it dark matter} Melvin Universe.
Categories: gr-qc hep-th