Reducible connections and non-local symmetries of the self-dual Yang-Mills equations

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.