Note on stationary-axisymmetric vacuum spacetimes with inside ellipsoidal symmetry

The author considers stationary-axisymmetric vacuum spacetimes possessing an additional 'inside' symmetry property; namely he assumes that the projected Riemannian 3-space induced by the stationary Killing field can be filled in with a one-parameter congruence of confocal ellipsoids. He presents a family of exact solutions of vacuum Einstein's equations possessing these symmetries, and it is shown that these spacetimes consist of a three-parameter family of solutions which coincide with the subfamily of the Kerr-NUT spacetimes with vanishing electric (and magnetic) charge. Hence, a simple geometrical characterization can be given for this subfamily of algebraically specialized Petrov type D solutions which resulted from the application of completely different spin coefficient techniques.