  We study the possibility of having Black hole of spherical and ring horizon
topology with five independent charges in the $U(1)^3$-model of 5D gauge
supergravity. To study these possibilities we consider not only the known
result obtained by local supersymmetry analysis but include the input coming
from non-local properties of the solutions, like the attractor mechanism, the
entropy function of Sen, the Euclidean formulation and general properties of
the uplift to ten dimension. For the spherical case, we found that there is no
room for more general Black holes than the ones already describe in
hep-th/0601156. On the other hand, if a solution of ring horizon topology
exists, we conclude that it must be labeled by three independent parameters
only, since it has to satisfy two independent constraints that we explicitly
find in terms of its chemical potentials. At the end of the article, based on
all the local and non-local information, we put forward a conjecture on the
constraints that characterize general Black holes dual to ${\cal N}=4$ SYM.
