Axial Obstructions to Rotating Bumblebee Vacuum Solutions

We identify two general obstructions to a regular rotating constant-norm vector vacua. Firstly, at the poles of a nondegenerate bifurcation surface, horizon and axial invariance force every smooth symmetry-inheriting one-form to vanish, contradicting a strictly nonzero constant norm. Secondly, we show that a conicity varying along a rotation axis generates a singular curvature that diverges as the inverse proper distance from the axis. We apply these results to a general three parameter family Kerr-disformal solution in Einstein–bumblebee gravity. Consequently, its vector vacuum is not smooth on the axes because of a nonzero constant norm, and the position-dependent conicity leads to genuine curvature singularities on the polar axes. A distinguished nonextremal branch nevertheless admits a regular outer Killing horizon away from the axes and defines an exact local rotating solution in the nonpolar region.