Integrability, Einstein spaces and holographic fluids

Using holographic-fluid techniques, we discuss some aspects of the integrability properties of Einstein's equations in asymptotically anti-de Sitter spacetimes. We review and we amend the results of 1506.04813 on how exact four-dimensional Einstein spacetimes, which are algebraically special with respect to Petrov's classification, can be reconstructed from boundary data: this is possible if the boundary metric supports a traceless, symmetric and conserved complex rank-two tensor, which is related to the boundary Cotton and energy-momentum tensors, and if the hydrodynamic congruence is shearless. We illustrate the method when the hydrodynamic congruence has vorticity and the boundary metric has two commuting isometries. This leads to the complete Plebanski-Demianski family. The structure of the boundary consistency conditions depict a U(1) invariance for the boundary data, which is reminiscent of a Geroch-like solution-generating pattern for the bulk.