Usman Kayani
We study supersymmetric near-horizon geometries of gauged D=5 supergravity coupled to vector multiplets, on the branch where the canonical rotational Killing vector $\tilde V$ of the horizon section $\mathcal{S}$ is non-vanishing. No rotational symmetry is assumed. On a compact, connected $\mathcal{S}$ without boundary, a second rotational Killing field, independent of $\tilde V$, always exists on the open set where $Z$ and $W$ are linearly independent, and extends to an isometry of all of $\mathcal{S}$, unconditionally if the moduli are constant and under a checkable extension hypothesis otherwise. With $K=Q_{IJ}C^IC^J$ vanishing exactly in the minimal theory, $K\equiv0$ recovers the known result of Grover, Gutowski, Papadopoulos and Sabra. When $K>0$, $P=Z+W$ vanishes at every critical point of $\|η_-\|^2$, and two sub-branches occur, both realised. On the first, $P\equiv0$, the moduli are constant and the isometry follows by homogeneity, recovering Kunduri and Lucietti local geometries as a consequence of supersymmetry and compactness rather than an ansatz. On the second, $P\not\equiv0$, the moduli genuinely vary; closure of the first-order system ($dP=0$) turns the Killing equation for a vector orthogonal to $P$ into a flat rank-two system transported along $P$, solved explicitly by $U=\|η_-\|^2(h_YN-h_NY)$. Both sub-branches are non-empty on compact $\mathcal{S}$: known $U(1)^3$ black holes populate the first, and an explicit compact horizon from the non-static Kunduri-Lucietti family, with $P$ vanishing at two points, populates the second and verifies the extension hypothesis numerically. We also prove $K$ is identically zero or everywhere positive, exclude a hidden Killing-tensor symmetry, and leave the $S^1\times S^2$ black-ring window $\tfrac13Φ^2<χ^2K\leq\tfrac43Φ^2$ unexcluded on either sub-branch.