On Waylen's Regular Axisymmetric Similarity Solutions

We review the similarity solutions proposed by Waylen for a regular time–dependent axisymmetric vacuum space–time, and show that the key equation introduced to solve the invariant surface conditions is related by a Bäcklund transform to a restriction on the similarity variables. We further show that the vacuum space–times produced via this path automatically possess a (possibly homothetic) Killing vector, which may be time–like.