  We review how Geroch's reduction method is extended from Ricci-flat to
Einstein spacetimes. The Ehlers-Geroch SL(2,R) group is still present in the
three-dimensional sigma-model that captures the dynamics, but only a subgroup
of it is solution-generating. Holography provides an alternative
three-dimensional perspective to integrability properties of Einstein's
equations in asymptotically anti-de Sitter spacetimes. These properties emerge
as conditions on the boundary data (metric and energy-momentum tensor) ensuring
that the hydrodynamic derivative expansion be resummed into an exact
four-dimensional Einstein geometry. The conditions at hand are invariant under
a set of transformations dubbed holographic U-duality group. The latter fills
the gap left by the Ehlers-Geroch group in Einstein spaces, and allows for
solution-generating maps mixing e.g. the mass and the nut charge.
