  We explain how to exploit systematically the structure of nilpotent orbits to
obtain a solvable system of equations describing extremal solutions of
(super-)gravity theories, i.e. systems that can be solved in a linear way. We
present the procedure in the case of the STU model, where we show that all
extremal solutions with a flat three-dimensional base are fully described with
the help of three different nilpotent orbits: the BPS, the almost-BPS and the
composite non-BPS. The latter describes a new class of solutions for which the
orientation of half of the constituent branes have been inverted with respect
to the BPS one, such that all the centres are intrinsically non-BPS, and
interact with each others. We finally recover explicitly the ensemble of the
almost-BPS solutions in our formalism and present an explicit two-centre
solution of the new class.
