Reductions by isometries of the self-dual Yang–Mills equations in four-dimensional Euclidean space

Abstract

A classification of the subgroups of the four-dimensional Euclidean group is given. The (anti-) self-dual Yang–Mills equations in four-dimensional Euclidean space are then reduced for each class representative and for any gauge group when a strict invariance of the gauge fields is required, or for the gauge group SO(3) if the gauge fields remain invariant up to gauge transformations. Amongst the residual systems, many constitute sets of first order ordinary differential equations sharing the Painlevé property.