  We construct the most general reducible connection that satisfies the
self-dual Yang-Mills equations on a simply connected, open subset of flat
$\mathbb{R}^4$. We show how all such connections lie in the orbit of the flat
connection on $\mathbb{R}^4$ under the action of non-local symmetries of the
self-dual Yang-Mills equations. Such connections fit naturally inside a larger
class of solutions to the self-dual Yang-Mills equations that are analogous to
harmonic maps of finite type.
