  We complete the study of static BPS, asymptotically AdS$_4$ black holes
within N=2 FI-gauged supergravity and where the scalar manifold is a
homogeneous very special Kahler manifold. We find the analytic form for the
general solution to the BPS equations, the horizon appears as a double root of
a particular quartic polynomial whereas in previous work this quartic
polynomial further factored into a pair of double roots. A new and
distinguishing feature of our solutions is that the phase of the supersymmetry
parameter varies throughout the black hole. The general solution has $2n_v$
independent parameters; there are two algebraic constraints on $2n_v+2$
charges, matching our previous analysis on BPS solutions of the form
$AdS_2\times \Sigma_g$. As a consequence we have proved that every BPS geometry
of this form can arise as the horizon geometry of a BPS AdS$_4$ black hole.
When specialized to the STU-model our solutions uplift to M-theory and describe
a stack of M2-branes wrapped on a Riemman surface in a Calabi-Yau fivefold with
internal angular momentum.
