Static spherically symmetric space-times with six Killing vectors

It had been proved earlier that spherically symmetric, static space-times have ten, seven, six, or four independent Killing vectors (KV's), but there are no cases in between. The case of six KV's is investigated here. It is shown that the space-time corresponds to a hyperboloid cross a sphere, reminiscent of Kaluza-Klein theory, with a compactification from four down to two dimensions. In effect, there is a unique metric for this space-time corresponding to a uniform mass distribution over all space.