with compact dimensions. In this case the space of static vacuum solutions, which need not have an isometry in the compact direction, is not analytically understood, and the solutions that have been found numerically so far have a rich structure [@Kol:2004ww; @Harmark:2005pp; @Harmark:2007md].[^1] A numerical approach exists in the literature to find cohomogenity-two static vacuum metrics, including black holes, employed first in the context of brane worlds [@star] and then Kaluza-Klein theory to find non-uniform and localized black holes [@Wiseman:2002zc; @Kudoh:2003ki; @Sorkin:2003ka]. A similar approach was used independently in four-dimensional Einstein-Yang-Mills systems in the earlier work of [@Kleihaus:1996vi; @Kleihaus:1997ic] (including extension to four-dimensional stationary systems [@Kleihaus:2000kg]) although the ‘constraint equations’ and their consistency was not discussed. The method has been used effectively in a variety of contexts [@Kudoh:2003xz; @Kudoh:2003vg; @Kudoh:2004kf; @Sorkin:2006wp; @Kleihaus:2006ee; @Kleihaus:2007cf; @Yoshino:2008rx; @Delsate:2009bd], but is fundamentally limited in that it requires cohomogenity two, a particular conformal coordinate system, and a certain ‘wizardry’ in actually making it work—for example, the ‘trick’ employed when treating a symmetry axis [@star; @Kol:2004ww].

In Kaluza-Klein theory, the static vacuum solution that preserves four-dimensional Poincaré invariance is trivial to write down analytically. However with additional compact dimensions even this is not true. For example, 