Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations

We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the $D$-dimensional vacuum Einstein equations with $D-2$ commuting Killing vector fields. A large class of exact solutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski–Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu–Sato) and five (single and double Myers–Perry black holes, black saturn, bicycling black rings).