  We study the problem of asymptotically flat bi-axially symmetric stationary
solutions of the vacuum Einstein equations in $5$-dimensional spacetime. In
this setting, the cross section of any connected component of the event horizon
is a prime $3$-manifold of positive Yamabe type, namely the $3$-sphere $S^3$,
the ring $S^1\times S^2$, or the lens space $L(p,q)$. The Einstein vacuum
equations reduce to an axially symmetric harmonic map with prescribed
singularities from $\mathbb{R}^3$ into the symmetric space
$SL(3,\mathbb{R})/SO(3)$. In this paper, we solve the problem for all possible
topologies, and in particular the first candidates for smooth vacuum
non-degenerate black lenses are produced. In addition, a generalization of this
result is given in which the spacetime is allowed to have orbifold
singularities. We also formulate conditions for the absence of conical
singularities which guarantee a physically relevant solution.
