  This article provides an explicit construction for a family of singular
instantons on S^4 S^2 with arbitrary real holonomy parameter \alpha. This
family includes the original \alpha = 1/4, c_2 = 3/2 solution discovered by P.
Forgacs, Z. Horvath, and L. Palla, and our approach is modeled on that of their
1981 paper. Our primary tool is the ansatz due to Corrigan, Fairlie, Wilczek,
and 't Hooft that constructs a self-dual Yang-Mills connection using a positive
real-valued harmonic super-potential. Here we reformulate this harmonic
function ansatz in terms of quaternionic notation, and we show that it arises
naturally from the Levi-Civita connection of a conformally Euclidean metric.
  To simplify the construction, we introduce an SO(3)-action on S^4, and we
show by dimensional reduction that the symmetric self-duality equation on S^4
is equivalent to the vortex equations over hyperbolic space H^2. We thus obtain
a similar harmonic function ansatz for hyperbolic vortices, which we also
derive using conformal transformations of H^2. Using this ansatz, we construct
the vortex equivalents of the symmetric 't Hooft instantons, and we prove using
the equivariant ADHM construction that they provide a complete description of
all hyperbolic vortices. We also analyze when two vortices constructed by this
ansatz are gauge equivalent, obtaining the surprising result that two such
vortices are completely determined by the gauge transformation between them.
