$ {\mathrm{AdS}}_{3}$ black holes with primary Proca hair from a regularized Gauss-Bonnet coupling

We construct a consistent three-dimensional Einstein-Gauss-Bonnet theory as a vector-tensor theory within the generalized Proca class by employing a regularization procedure based on the Weyl geometry, which was introduced recently by Charmousis, Fernandes, and Hassaine [Phys. Rev. D 111, 124008 (2025).]. We then obtain an asymptotically Anti-de Sitter ($ {\text{AdS}}_{3}$), static, and circularly symmetric black hole solution with primary Proca hair. Afterward, we investigate the effect of the scalar-tensor Gauss-Bonnet coupling constructed previously by different regularization schemes. We further generalize these solutions by incorporating an electric charge. As special cases, we find a regular black hole solution in addition to charged and uncharged stealth Banados-Teitelboim-Zanelli black hole solutions.