Benenti-Francaviglia separability and Petrov types

Benenti and Francavilla (BF) proposed a class of metrics with two commuting Killing vectors for which there exists an irreducible Killing tensor of the second rank and the geodesic equations are integrable. This class admits a non-trivial Ricci tensor and generically is not algebraically special. We find an additional condition on the BF class, under which the metrics admit isotropic geodesic and shear-free congruences, and belong either to the general Petrov type I or to type D, but not to type II. The corresponding Killing tensors have only two nonzero Newman-Penrose projections. This subclass includes black holes with the Newman-Unti-Tamburino (NUT) parameter, in the = 4 supergravity.