On the construction of exact solutions of the stationary axially symmetric Einstein equations

Summary For the stationary axially symmetric vacuum, field, the Einstein equations reduce to a system for the logarithmic derivatives of the unknown function. An inverse scattering formula and a Bäcklund transformation are presented for the reduced system. As the simplest case, successive uses of the Bäcklund transformation give the Kasner type solutions and their generalization starting from the Minkowski metric.