  We study the problem of stationary bi-axially symmetric solutions of the
$5$-dimensional minimal supergravity equations. Essentially all possible
solutions with nondegenerate horizons are produced, having the allowed horizon
cross-sectional topologies of the sphere $S^3$, ring $S^1\times S^2$, and lens
$L(p,q)$, as well as the three different types of asymptotics. The solutions
are smooth apart from possible conical singularities at the fixed point sets of
the axial symmetry. This analysis also includes the solutions known as solitons
in which horizons are not present but are rather replaced by nontrivial
topology called bubbles which are sustained by dipole fluxes. Uniqueness
results are also presented which show that the solutions are completely
determined by their angular momenta, electric and dipole charges, and rod
structure which fixes the topology. Consequently we are able to identify the
finite number of parameters that govern a solution. In addition, a
generalization of these results is given where the spacetime is allowed to have
orbifold singularities.
