Modified Kerr black holes surrounded by dark matter spike

We study supermassive black holes (SMBH), surrounded by a dark matter (DM) spike, that can be found at the centers of Milky Way and $\text{M87}$ galaxies and are accompanied by a specific kind of topological defect. The investigation is developed within the framework of Bumblebee Gravity with a global monopole (BGGM). The dark matter spike is described by a power-law density profile. Our main objective is to assess how the background arising from spontaneous Lorentz symmetry breaking and the presence of a global monopole influence the properties of the Kerr BH within the region affected by the spike. Using a spherically symmetric static BH with BGGM properties as the seed metric, we construct a non-rotating spacetime with a DM spike, resulting in a BGGM-motivated Schwarzschild-like BH by solving the modified Tolman–Oppenheimer–Volkoff equations (TOV). Next, we extend this approach to the case of a rotating spacetime resulting in the BGGM-motivated Kerr-like BH (BGMKLBH). This approach allows us to explore the spacetime structure, and the BGMKLBH shadows. Then, using available observational data for the DM spike density and considering the effects of BGGM on ${\text{Sgr A}}^{\ast }$ and ${\text{M87}}^{\ast }$ SMBHs, we analyze the shapes of their shadows and put constraints on the BGGM parameter. Thus, we infer that the BGMKLBHs could be reliable candidates for the astrophysical BHs.