in a modern context are given weight by string theory and other phenomenological and formal considerations, such as Randall-Sundrum and large-extra-dimensions scenarios [@ADD; @ADD2; @Randall:1999ee; @Randall:1999vf] and the AdS-CFT correspondence [@AdSCFTreview]. However the space of static and stationary solutions is considerably more complicated in these contexts, and many solutions known or believed to exist currently have no analytic solution.

For asymptotically flat $D$-dimensional gravity in the absence of matter, with $D>4$, static black holes are given uniquely by the Schwarzschild-Tangherlini black hole [@Tangherlini:1963bw; @Gibbons:2002av; @Gibbons:2002bh]. However, for stationary solutions in $D = 5$, in addition to the Myers-Perry solution which generalizes Kerr [@MP], the Emparan-Reall black ring is found [@Ring]; these may even be combined into a ‘black Saturn’ [@Elvang:2007rd; @Elvang:2007hg]. Whilst these are understood analytically, in $D > 5$ the black ring solution is not known and it is thought that the structure of stationary solutions is yet more complicated than in $D = 5$ [@Emparan:2003sy; @Emparan:2007wm], with little hope of admitting closed-form solutions.

Now consider Kaluza-Klein theory, by which we mean gravity in five dimensions such that the asymptotic geometry is a product of four-dimensional Minkowski space and a circle. This is the canonical toy model for theories 