Black holes surrounded by Einstein clusters as models of dark matter
  fluid
  We construct a novel class of spherically symmetric and asymptotically flat
black holes and naked singularities surrounded by anisotropic dark matter fluid
with the equation of state (EoS) of the form $P_t=\omega \rho$. We assume that
dark matter is made of weakly interacting particles orbiting around the
supermassive black hole in the galactic center and the dark matter halo is
formed by means of Einstein clusters having only tangential pressure. In the
large distance from the black hole we obtain the constant flat curve with the
upper bound for the dark matter state parameter $\omega\lsim 10^{-7}$. To this
end, we also check the energy conditions of the dark matter fluid outside the
black hole/naked singularity, the deflection of light by the galaxy, and the
shadow images of the Sgr A$^\star$ black hole using the rotating and radiating
particles. For the black hole case, we find that the effect of dark matter
fluid on the shadow radius is small, in particular the angular radius of the
black hole is shown to increase by the order $10^{-4} \mu$arcsec compared to
the vacuum solution. For the naked singularity we obtain significantly smaller
shadow radius compared to the black hole case. Finally, we study the stability
of the S2 star orbit around Sgr A$^\star$ black hole under dark matter effects.
It is argued that the motion of S2 star orbit is stable for values $\omega
\lsim 10^{-7}$, however further increase of $\omega $ leads to unstable orbits.
