Asymptotically flat solutions to the Ernst equation with reflection symmetry

It is shown that the class of asymptotically flat solutions to the axisymmetric and stationary vacuum Einstein equations with reflection symmetry of the metric is uniquely characterized by a simple relation for the Ernst potential on the upper part of the symmetry axis ( axis): . This result generalizes a well-known fact from potential theory: axisymmetric solutions to the Laplace equation that vanish at infinity and have reflection symmetry with respect to the plane are characterized by a potential that is an odd function of on the upper part of the axis.