The uniqueness problem of static axially symmetric black hole in a magnetic universe filled with dark matter component is considered in this paper. The dark matter model comprises of the additional $U(1)$ gauge field (dark photon) interacting with the Maxwell one through the kinetic mixing term. We show that all the solutions of Einstein-Maxwell dark photon gravity subject to the same boundary and regularity conditions authorize the only static axially symmetric black hole solutions with nonvanishing time and azimuthal components of Maxwell and hidden-sector gauge fields, e.g., Schwarzschild-like black hole immersed in a dark matter Melvin universe.

• Received 20 June 2023
• Accepted 29 August 2023

DOI:https://doi.org/10.1103/PhysRevD.108.064026