  The continuous point symmetry algebra of the hyperbolic Ernst equation is
presented. In a second step the corresponding group transformations are
considered. Accordingly, the solutions of the hyperbolic Ernst equation that
are invariant under Lie point symmetries, are constructed from the related
invariant surface conditions. Furthermore, all these solutions are revealed to
be related to solutions of the Euler-Poisson-Darboux equation by a simple
coordinate transformation. The parallels of these results to coordinate
transformations, which are important in the context of colliding plane wave
space times, are pointed out.
