The full integration of black hole solutions to symmetric supergravity theories

We prove that all stationary and spherical symmetric black hole solutions to theories with symmetric target spaces are integrable and we provide an explicit integration method. This exact integration is based on the description of black hole solutions as geodesic curves on the moduli space of the theory when reduced over the time-like direction. These geodesic equations of motion can be rewritten as a specific Lax pair equation for which mathematicians have provided the integration algorithms when the initial conditions are described by a diagonalizable Lax matrix. On the other hand, solutions described by nilpotent Lax matrices, which originate from extremal regular (small) D=4D=4 black holes can be obtained as suitable limits of solutions obtained in the diagonalizable case, as we show on the generating geodesic   (i.e. most general geodesic modulo global symmetries of the D=3D=3 model) corresponding to regular (and small) D=4D=4 black holes. As a byproduct of our analysis we give the explicit form of the “Wick rotation” connecting the orbits of BPS and non-BPS solutions in maximally supersymmetric supergravity and its STU truncation.