  The galactic black hole is a supermassive black hole located at the center of
a galaxy surrounded by a dark matter halo. For the first time, we establish a
generic, fully-relativistic formalism to calculate solutions of Einstein's
gravity minimally coupled to an anisotropic fluid modeled by the Einstein
cluster in axisymmetric, non-vacuum spacetimes, which are extensions of
spherically-symmetric cases. These asymptotically flat spacetimes with regular
horizons can describe the geometry of galaxies harboring supermassive black
holes and are useful to constrain the environment surrounding astrophysical
black holes. Our findings provide a solid groundwork for future studies on the
shadows, quasi-normal modes, and other phenomena associated with rotating
galactic black holes.
