  We consider the octonionic self-duality equations on eight-dimensional
manifolds of the form $M_8=M_4\times \R^4$, where $M_4$ is a hyper-K\"ahler
four-manifold. We construct explicit solutions to these equations and their
symmetry reductions to the non-abelian Seiberg-Witten equations on $M_4$ in the
case when the gauge group is SU(2). These solutions are singular for flat and
Eguchi-Hanson backgrounds. For $M_4=\R\times {\mathcal G}$ with a cohomogeneity
one hyper-K\"ahler metric, where ${\mathcal G}$ is a nilpotent (Bianchi II) Lie
group, we find a solution which is singular only on a single-sided domain wall.
This gives rise to a regular solution of the non-abelian Seiberg-Witten
equations on a four-dimensional nilpotent Lie group which carries a regular
conformally hyper-K\"ahler metric.
