Moduli space of stationary vacuum black holes from integrability

We consider the classification of asymptotically flat, stationary, vacuum black hole spacetimes in four and five dimensions, that admit one two commuting axial Killing fields respectively. It is well known Einstein equations reduce to a harmonic map on two-dimensional orbit space, which itself arises as integrability condition for linear system spectral equations. integrate Belinski-Zakharov along boundary space use this fully determine metric associated Ernst twist potentials axes horizons. This sufficient derive moduli solutions are free conical singularities axes, any given rod structure. As an illustration method we obtain constructive uniqueness proofs Kerr Myers-Perry holes doubly spinning rings.