  Explicit solutions to the non-linear field equations of some gravitational
theories can be obtained, by means of a Riemann-Hilbert approach, from a
canonical Wiener-Hopf factorisation of certain matrix functions called
monodromy matrices. In this paper we describe other types of factorisation from
which solutions can be constructed in a similar way. Our approach is based on
an invariance problem, which does not constitute a Riemann-Hilbert problem and
allows to construct solutions that could not have been obtained by Wiener-Hopf
factorisation of a monodromy matrix. It gives rise to a novel solution
generating method based on matrix multiplications. We show, in particular, that
new solutions can be obtained by multiplicative deformation of the canonical
Wiener-Hopf factorisation, provided the latter exists, and that one can
superpose such solutions. Examples of applications include Kasner,
Einstein-Rosen wave and gravitational pulse wave solutions.
