Petrov types, separability, and generalized photon surfaces of supergravity black holes

The vacuum and electrovacuum Einstein equations for spacetimes with two commuting Killing vectors can be solved by indirect methods of integrable systems. But if, in addition, the spacetime admits an irreducible Killing tensor and the corresponding Klein-Gordon equation is separable, they can be integrated directly by separation of variables, as shown by Carter in 1968. We generalize this approach to supergravity and derive a metric ansatz that ensures the above properties for Petrov-type $I$. Our derivation is based on the Benenti-Francavilla ansatz for metrics admitting two commuting Killing vectors and an irreducible Killing tensor. We find additional constraints that guarantee the existence of two shear-free null geodesic congruences and the separability of the Klein-Gordon equation. The resulting class of metrics belongs to a certain sector of Petrov type $I$, called ${I}_{B}$, whose algebraically special subsector contains only type $D$. For this class, a direct integration of the supergravity equations seems possible. We also show that these spacetimes admit a general description of the generalized photon and massive particle surfaces recently introduced in connection with black hole shadows.