Vacuum solutions of five dimensional Einstein equations generated byinverse scattering method. II. Production of the black ring solution

We study vacuum solutions of five-dimensional Einstein equations generatedby the inverse scattering method. We reproduce the black ring solution whichwas found by Emparan and Reall by taking the Euclidean Levi-Civit\`ametric plus one-dimensional flat space as a seed. This transformation consistsof two successive processes; the first step is to perform the three-solitonictransformation of the Euclidean Levi-Civit\`a metric with one-dimensionalflat space as a seed. The resulting metric is the Euclidean C-metricwith extra one-dimensional flat space. The second is to perform the two-solitonictransformation by taking it as a new seed. Our result may serve as a steppingstone to find new exact solutions in higher dimensions.