  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).
