Ricci flat metrics with bidimensional null orbits and non-integrable orthogonal distribution (Q997143)

In the present article the author finds all metrics \(g\) in a \(4\)-dimensional manifold of any signature with vanishing Ricci tensor, that satisfy three conditions: (i) \(g\) allows a Lie algebra of Killing vector fields with \(2\)-dimensional orbits, (ii) the tensor \(g\) degenerates when restricted to the orbits, and (iii) the distribution orthogonal to the orbits is not integrable. The main result is that there exists only one (up to sign), Ricci flat metric.