can be overcome either by using Ricci-flow and tuning initial data, or by using Newton’s method. In the Lorentzian context, solutions may be found independently of their dynamical stability properties. Whilst these methods do not assume any isometries, if such isometries exist, it is natural to exploit them to effectively reduce the dimensionality of the problem.

It is instructive to see the two methods described in Section \[sec:algorithms\] demonstrated in a non-trivial example. Thus in Section \[sec:KK\] we test both these methods by finding the various static black holes in five-dimensional Kaluza-Klein theory—the ‘localized black holes’ and ‘non-uniform black strings’. We find that Newton’s method gives excellent results in practice and is considerably easier to use than the Ricci-flow method with its tuning of initial data. We compute various properties of these black holes, such are their thermodynamic properties and horizon geometry. We are able to provide good evidence for a merger between the localized and non-uniform solutions [@Kol:2004ww; @Harmark:2005pp; @Harmark:2007md], computing the localized branch closer to the potential merger point than previously done in [@Kudoh:2004hs] even though we have used more modest resources.

New results include finding that the localized solutions have a minimum temperature which divides the localized 