Kerr-Schild transformation of the Benenti-Francaviglia metric

The Benenti-Francaviglia (BF) family of metrics provides the most general form of a spacetime metric that admits two mutually commuting Killing vectors and an irreducible Killing tensor. The geodesic equations for the BF family are thus completely integrable by separation of variables. Within this broad class, we explore the Kerr-Schild transformation of a degenerate subclass distinguished by the existence of a shear-free null geodesic congruence. By requiring the deformed metric to preserve the Killing symmetry and circularity, we demonstrate that the deformed metric again falls into the degenerate BF family, modulo the replacement of a single structure function. We apply the present algorithm to ${\cal N}=2$ gauged supergravity and obtain a dyonic generalization of the Chong-Cvetič-Lü-Pope rotating black hole solution, by taking the background metric to be a solution of the Einstein-scalar gravity. The present prescription extends to five dimensions, provided that the constant of geodesic motion associated with the extra Killing direction vanishes. The same reasoning applies to the case where the background degenerate BF metric is distorted in a (non)conformal manner. Our formalism offers a unified perspective on the relation between seed and deformed metrics in the Kerr-Schild construction.