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4D gravity is not fully integrable. Nevertheless, one may hope that certain sectors are. In this spirit, we consider gravitational asymptotic symmetries, namely transformations of spacetime coordinates that leave the large-scale behavior of the gravitational field invariant. Such transformations are described by the so-called BMS group. While subtleties do persist in four dimensions, the three-dimensional case offers an appealing playground. There, we uncover the presence of an integrable hierarchy.