  We study "minimal degree" complete bases of duality- and "horizontal"-
invariant homogeneous polynomials in the flux representation of two-centered
black hole solutions in two classes of D=4 Einstein supergravity models with
symmetric vector multiplets' scalar manifolds. Both classes exhibit an SL(2,R)
"horizontal" symmetry. The first class encompasses N=2 and N=4 matter-coupled
theories, with semi-simple U-duality given by SL(2,R) x SO(m,n); the analysis
is carried out in the so-called Calabi-Vesentini symplectic frame (exhibiting
maximal manifest covariance) and until order six in the fluxes included. The
second class, exhibiting a non-trivial "horizontal" stabilizer SO(2), includes
N=2 minimally coupled and N=3 matter coupled theories, with U-duality given by
the pseudo-unitary group U(r,s) (related to complex flux representations).
Finally, we comment on the formulation of special Kaehler geometry in terms of
"generalized" groups of type E7.
