SU(2)SU(2) solutions to self-duality equations in eight dimensions

We consider the octonionic self-duality equations on eight-dimensional manifolds of the form M8=M4×R4M8=M4×R4, where M4M4 is a hyper-Kähler four-manifold. We construct explicit solutions to these equations and their symmetry reductions to the non-abelian Seiberg–Witten equations on M4M4 in the case when the gauge group is SU(2)SU(2). These solutions are singular for flat and Eguchi–Hanson backgrounds. For M4=R×GM4=R×G with a cohomogeneity one hyper-Kähler metric, where GG 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-Kähler metric.