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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 in- variant 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

Geroch group for Einstein spaces and holographic integrability