  The non-autonomous chiral model equation for an $m \times m$ matrix function
on a two-dimensional space appears in particular in general relativity, where
for $m=2$ a certain reduction of it determines stationary, axially symmetric
solutions of Einstein's vacuum equations, and for $m=3$ solutions of the
Einstein-Maxwell equations. Using a very simple and general result of the
bidifferential calculus approach to integrable partial differential and
difference equations, we generate a large class of exact solutions of this
chiral model. The solutions are parametrized by a set of matrices, the size of
which can be arbitrarily large. The matrices are subject to a Sylvester
equation that has to be solved and generically admits a unique solution. By
imposing the aforementioned reductions on the matrix data, we recover the Ernst
potentials of multi-Kerr-NUT and multi-Demianski-Newman metrics.
