Group Transformation That Generates the Kerr and Tomimatsu-Sato Metrics

For stationary axially symmetric vacuum metrics, we give a series of transformations ${\beta }^{\mathrm{}\left(k\right)\mathrm{}}$ which automatically preserve asymptotic flatness. We show how to generate the Kerr metric from the Schwarzschild, using ${\beta }^{\mathrm{}\left(0\right)\mathrm{}}$. We also show, using ${\beta }^{\mathrm{}\left(k\right)\mathrm{}}$, that the Tomimatsu-Sata (TS) class of metrices must be larger than previously realized, and for $\delta =2\mathrm{}$ there is a five-parameter TS metric. As an example, we present a two-parameter metric from this family, which we claim to be a new, physically realistic, asymptotically flat, rotating vacuum solution.