Symmetries of stationary axially symmetric vacuum Einstein equations and the new family of exact solutions

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Abstract
A new family of exact solutions of the stationary axially symmetric vacuum Einstein equations is presented. The internal symmetries, SL(2,R) rotation, and duality of parametrization, are combined to construct a Bäcklund transformation. For the special ansatz of the field equations, the Bäcklund transformation can be integrated and the hierarchy of ansatz is generated, recursively. The Riemann–Hilbert problem is also discussed for the inverse scattering formulas generating nonlocal symmetries.