  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 = 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 = 3 model) corresponding to regular
(and small) D = 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.
