This paper is devoted to the studies of the similarity solutions due to P. C. Waylen [Proc. R. Soc. Lond., Ser. A 411, 49-57 (1987; Zbl 0618.53063); ibid. A 440, 711-715 (1993; Zbl 0792.53072)] for a regular time dependent axisymmetric vacuum space-time. The authors describe Waylen’s exact similarity solution results for an axisymmetric vacuum space-time and analyze the point symmetries in the similarity solution with special attention to the boundary conditions so as to derive the conditions for the similarity variables. In the course of this discussion, it turns out that the point symmetries form an infinite-dimensional Lie pseudogroup. The authors show that all similarity solutions possess an additional Killing (possibly homothetic) vector which may be time-like, for the space-time metric. Finally, once having known the point symmetries of the field equations and the invariance conditions for similarity variables, the authors then establish the link between the invariance conditions and Waylen’s equation as a Bäcklund map which, in fact, represents a restriction on the set of possible similarity solutions.