  We study the integrability of gravity-matter systems in D=2 spatial
dimensions with matter related to a symmetric space G/K using the well-known
linear systems of Belinski-Zakharov (BZ) and Breitenlohner-Maison (BM). The
linear system of BM makes the group structure of the Geroch group manifest and
we analyse the relation of this group structure to the inverse scattering
method of the BZ approach in general. Concrete solution generating methods are
exhibited in the BM approach in the so-called soliton transformation sector
where the analysis becomes purely algebraic. As a novel example we construct
the Kerr-NUT solution by solving the appropriate purely algebraic
Riemann-Hilbert problem in the BM approach.
