  We consider again stationary solutions to the spherically symmetric
Einstein--Maxwell--Klein--Gordon system, commonly known as ``charged boson
stars'', originally studied by Jetzer and Van Der Bij. We construct families of
charged boson stars in the ground state, for different values of the charge
parameter $q$, and different values of the central scalar field. Following
Jetzer and Van Der Bij, one can define a critical value for the charge $q=q_c$
that corresponds to the value for which the Coulomb repulsion of the bosonic
particles exactly cancels their newtonian gravitational attraction. We confirm
the claim made by Pugliese et al. that super-critical solutions exist for a
limited range of charges above the critical value $q>q_c$ (though we find an
even smaller range of $q$ for which this is possible). Our analysis indicates,
however, that all such super-critical solutions are gravitationally unbound,
and are therefore expected to be unstable. One of the main results of our
analysis is the fact that, even though we do find a family of slightly
super-critical solutions in the sense that $q>q_c$, there are no super-critical
solutions in the sense that the total charge $Q$ is larger than the total mass
$M$ of the system.
