  We investigate supermassive black holes (SMBH), surrounded by a dark matter
(DM) spike, that can be found at the centers of Milky Way and M87 galaxies and
are accompanied by a specific kind of topological defect. The investigation is
developed within the framework of Bumblebee Gravity (BG) with a global monopole
(GM). The power-law density profile, describing the DM spike, is expected to
follow a power-law distribution starting from a few times the BH horizon radius
and extending outward to an external radius Rsp. We determine the normalization
parameter, \r{ho}0, for the density profile to ensure its compatibility with
both the galaxy total mass and the enclosed DM mass within the radius of
influence of Sagittarius A (Sgr A) and M87 SMBHs, approximately of order 105Rs.
Our main objective is to assess how the background arising from spontaneous
Lorentz symmetry (LS) breaking and the presence of a GM influence the
properties of the Kerr BH within the region affected by the spike. The scarcity
of non-rotating and rotating BH models embedded in a DM spike and subject to BG
and GM (BGGM) effects significantly hinders the progress in testing BGGM
through observations. To address this issue, we initiate our investigation by
considering a spherically symmetric static BH with BGGM properties as the seed
metric. From there, we construct a non-rotating spacetime with a DM spike
(similar to the Schwarzschild spacetime), leading to the emergence of the
BGGM-motivated Schwarzschild-like BH (BGMSLBH). To construct the BGMSLBH
spacetime, we employ two different methods: one involves solving the modified
Tolman-Oppenheimer-Volkoff (TOV) equations, while the other entails solving the
modified Einstein field equations. In the latter approach, we consider the
energy-momentum tensor incorporating the contributions of DM spike and BH
counterpart of the BGGM.
