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Solutions associated with the point symmetries of the hyperbolic Ernst equation

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.