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In this paper we apply in a systematic way a previously developed integrationalgorithm of the relevant Lax equation to the construction of sphericalsymmetric, asymptotically flat black hole solutions of N=2 supergravities withsymmetric Special Geometry. Our main goal is the classification of theseblack-holes according to the H*-orbits in which the space of possible Laxoperators decomposes, H* being the isotropy group of scalar manifoldoriginating from time-like dimensional reduction of supergravity from D=4 toD=3 dimensions. The main result of our investigation is the construction ofthree universal tensors, extracted from quadratic and quartic powers of the Laxoperator, that are capable of classifying both regular and nilpotent H* orbitsof Lax operators. Our tensor based classification is compared, in the case ofthe simple one-field model S^3, to the algebraic classification of nilpotentorbits and it is shown to provide a simple and practical discriminating method.We present a detailed analysis of the S^3 model and its black hole solutions,discussing the Liouville integrability of the corresponding dynamical system.By means of the Kostant-representation of a generic Lie algebra element, wewere able to develop an algorithm which produces the necessary number ofhamiltonians in involution required by Liouville integrability of genericorbits. The degenerate orbits correspond to extremal black-holes and arenilpotent. We analyze these orbits in some detail working out differentrepresentatives thereof and showing that the relation between H* orbits andcritical points of the geodesic potential is not one-to-one. Finally we presentthe conjecture that our newly identified tensor classifiers are universal andable to label all regular and nilpotent orbits in all homogeneous symmetricSpecial Geometries.

Integrability of Supergravity Black Holes and New Tensor Classifiers of Regular and Nilpotent Orbits