  We describe the systematical construction of the first order formalism for
multi-centered black holes with flat three dimensional base-space, within the
so-called $T^{3}$ model of N=2, D=4 ungauged Maxwell-Einstein supergravity. The
three possible flow classes (BPS, composite non-BPS and almost-BPS) are
analyzed in detail, and various solutions, such as single-centered (static or
under-rotating) and all known multi-centered black holes, are recovered in this
unified framework. We also consider the possibility of obtaining new solutions.
The almost-BPS class is proved to split into two general sub-classes,
corresponding to a positive or negative value of the duality-invariant
polynomial for the total charge; the well known almost BPS system is shown to
be a particular solution of the second sub-class.
