  We present a classification of asymptotically flat, supersymmetric black hole
and soliton solutions of five-dimensional minimal supergravity that admit a
single axial symmetry which `commutes' with the supersymmetry. This includes
the first examples of five-dimensional black hole solutions with exactly one
axial Killing field that are smooth on and outside the horizon. The solutions
have similar properties to the previously studied class with biaxial symmetry,
in particular, they have a Gibbons-Hawking base and the harmonic functions must
be of multi-centred type with the centres corresponding to the connected
components of the horizon or fixed points of the axial symmetry. We find a
large moduli space of black hole and soliton spacetimes with non-contractible
2-cycles and the horizon topologies are $S^3$, $S^1\times S^2$ and lens spaces
$L(p,1)$.
