  Non-compact hyperk\"ahler spaces arise frequently in gauge theory. The
4-dimensional hyperk\"ahler ALE spaces are a special class of non-compact
hyperk\"ahler spaces. They are in one-to-one correspondence with the finite
subgroups of SU(2) and have interesting connections with representation theory
and singularity theory, captured by the McKay Correspondence.
  The 4-dimensional hyperk\"ahler ALE spaces are first classified by Peter
Kronheimer via a finite-dimensional hyperk\"ahler reduction. In this paper, we
give a new gauge-theoretic construction of these spaces. More specifically, we
realize each 4-dimensional hyperk\"ahler ALE space as a moduli space of
solutions to a system of equations for a pair consisting of a connection and a
section of a vector bundle over an orbifold Riemann surface, modulo a gauge
group action. The construction given in this paper parallels Kronheimer's
original construction and hence can also be thought of as a gauge-theoretic
interpretation of Kronheimer's construction of these spaces.
