  The correspondence between stationary, axisymmetric, asymptotically flat
space-times and bundles over a reduced twistor space has been established in
four dimensions. The main impediment for an application of this correspondence
to examples in higher dimensions has been the lack of a higher-dimensional
equivalent of the Ernst potential. This article will propose such a generalized
Ernst potential, point out where the rod structure of the space-time can be
found in the twistor picture and thereby provide a procedure for generating
solutions to the Einstein equations in higher dimensions from the rod structure
and other asymptotic data. An important result for the study of
five-dimensional examples will be the theorem which relates the patching
matrices on the outer semi-infinite rods.
