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 generated by the inverse scattering method. We reproduce the black ring solution which was found by Emparan and Reall by taking the Euclidean Levi-Cività metric plus one-dimensional flat space as a seed. This transformation consists of two successive processes; the first step is to perform the three-solitonic transformation of the Euclidean Levi-Cività metric with one-dimensional flat space as a seed. The resulting metric is the Euclidean $C$-metric with extra one-dimensional flat space. The second is to perform the two-solitonic transformation by taking it as a new seed. Our result may serve as a stepping stone to find new exact solutions in higher dimensions.