Regular $(2+1)$-dimensional spatially homogeneous ${\alpha }^{\mathrm{\prime }}$-corrected BTZ-like black hole in string theory

We consider a $(2+1)$-dimensional spacetime whose two-dimensional space part is Weyl-related to a surface of arbitrary negative constant Gaussian curvature with symmetries of two-dimensional Lie algebra. It is shown that the geometry is a Lobachevsky-type geometry described by deformed hyperbolic function. At leading order string effective action with the source given by dilaton and antisymmetric $B$-field in the presence of central charge deficit term $\mathrm{\Lambda }$, we obtained a solution whose line element is Weyl-related to this homogeneous spacetime with arbitrary negative Gaussian curvature. The solution can be transformed to the BTZ-like black hole by coordinate redefinition, while the BTZ black hole can be recovered by choosing a special set of parameters. The solutions appear to be in the high curvature limit $R{\alpha }^{\mathrm{\prime }}\gtrsim 1$, with emphasis on including the higher order ${\alpha }^{\mathrm{\prime }}$ corrections. Considering the two-loop (first order ${\alpha }^{\mathrm{\prime }}$) $\beta$-function equations of $\sigma$-model, we also present the ${\alpha }^{\mathrm{\prime }}$-corrected black hole solutions.