  Compact boson stars, whose scalar field vanishes identically in the exterior
region, arise in a theory involving a {\it massless} complex scalar field with
a conical potential, when coupled to gravity. Their charged compact
generalizations, obtained in the presence of a U(1) gauge field, exhibit
further interesting features. On the one hand, charged compact boson shells can
arise, whose scalar field vanishes also in the central region, while on the
other hand, the domain of existence of charged compact boson stars exhibits
bifurcation points. First 2D phase diagrams have been studied before. Here we
extend these earlier studies to a larger range of the variables and study
additional phase diagrams. We then extend these studies to obtain 3D phase
diagrams and present these with a detailed discussion of their various regions
with respect to the bifurcation points and argue, that there is an infinite
series of such bifurcation points. Thus the theory is seen to contain rich
physics in a particular domain of the phase diagrams. We also discuss the
dependence of the fields on the dimensionless radial coordinate for some
representative points of the phase trajectories in the phase diagrams of the
theory.
