Global structure of static Euclidean SU(2) solutions

I investigate the structure of static Euclidean SU(2) fields by using both explicit solutions and topological and variational arguments. I characterize the general $\mathrm{}\left|n\right|=1\mathrm{}$ finite-energy static SU(2) field by the set of points (the zero-set) on which ${\overrightarrow{\mathrm{b}}}^0$ vanishes, and argue that the Prasad-Sommerfield solution, which has an isolated point zero-set, is in fact the degenerate limit of a much wider class of (anti-) self-dual distribution solutions with 1-, 2-, or 3-dimensional zero-sets. In particular, I give arguments suggesting that there are (anti-) self-dual "string" configurations with a line segment as zero-set, and that these solve the source-free static Euclidean field equations. The possible role of such solutions as background fields in the quarkconfinement problem is discussed, and a program of numerical investigations is outlined.