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Stationary black holes of massless supergravity theories are described bycertain geodesic curves on the target space that is obtained after dimensionalreduction over time. When the target space is a symmetric coset space we makeuse of the group-theoretical structure to prove that the second order geodesicequations are integrable in the sense of Liouville, by explicitly constructingthe correct amount of Hamiltonians in involution. This implies that theHamilton-Jacobi formalism can be applied, which proves that all such black holesolutions, including non-extremal solutions, possess a description in terms ofa (fake) superpotential. Furthermore, we improve the existing integrationmethod by the construction of a Lax integration algorithm that integrates thesecond order equations in one step instead of the usual two step procedure. Weillustrate this technology with a specific example.

Black holes in supergravity and integrability