  We consider the dimensional reduction to two dimensions of certain
gravitational theories in $D \geq 4$ dimensions at the two-derivative level. It
is known that the resulting field equations describe an integrable system in
two dimensions which can also be obtained by a dimensional reduction of the
self-dual Yang-Mills equations in four dimensions. We use this relation to
construct a single copy prescription for classes of gravitational solutions in
Weyl-Lewis-Papapetrou coordinates. In contrast with previous proposals, we find
that the gauge group of the Yang-Mills single copy carries non-trivial
information about the gravitational solution. We illustrate our single copy
prescription with various examples that include the extremal Reissner-Nordstrom
solution, the Kaluza-Klein rotating attractor solution, the Einstein-Rosen wave
solution and the self-dual Kleinian Taub-NUT solution.
