  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.
