In this work, we show that if the bumblebee field in the Einstein-bumblebee theory is given by its vacuum expectation value ($B_{\mu }=b_{\mu }$) and it is not dynamical (${\partial }_{\mu }{B}_{\nu }-{\partial }_{\nu }{B}_{\mu }=0$), then these conditions uniquely provide a generating technique, allowing us to construct exact solutions to bumblebee gravity from the vacuum solutions by adding a term $\sim b_{\mu }{b}_{\nu }$ to the metric tensor (thus proving the uniqueness of the method, presented in Poulis and Soares [Exact modifications on a vacuum spacetime due to a gradient bumblebee field at its vacuum expectation value, Eur. Phys. J. C 82, 613 (2022)]). Also, we show that the bumblebee field within this technique is proportional to the tangential vector of the (timelike or spacelike) geodesic curve in the background vacuum spacetime, and can be easily found knowing the solution to the Hamilton-Jacobi equation. Moreover, we prove that this technique can be extended to the case of any nonzero cosmological constant and the presence of the electromagnetic field. We apply this generating technique and obtain the bumblebee extension of the Kerr-Newman-Taub-NUT-(anti–)de Sitter spacetime. We show that this extension is not unique, as it depends on the exact geodesic curve one chooses to associate a bumblebee field with. Also, we show that for some choices of parameters, the bumblebee field and the metric functions may become complex, and to avoid this, we find the conditions under which the bumblebee field remains globally real.

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• Alternative gravity theories
• Classical black holes
• General relativity