  We prove existence of all possible bi-axisymmetric near-horizon geometries of
5-dimensional minimal supergravity. These solutions possess the cross-sectional
horizon topology $S^3$, $S^1\times S^2$, or $L(p,q)$ and come with prescribed
electric charge, two angular momenta, and a dipole charge (in the ring case).
Moreover, we establish uniqueness of these solutions up to an isometry of the
symmetric space $G_{2(2)}/SO(4)$.
