  Considering the possibility to build an $e^+e^-$ collider at the energies
around $Z$-boson resonance with a luminosity so high as ${\cal L} \propto
10^{34}cm^{-2}s^{-1}$ (even higher) and the abilities of a modern synthesis
detector, we systematically calculate the exclusive two body processes:
$e^+e^-$ annihilates into a heavy quarkonium and a photon (initiate state
radiation i.e. ISR is involved), at the energies around the $Z$-boson
resonance. Since the couplings of $Z$-boson to quarks contain axial vector as
well as vector, so here the produced heavy quarkonium may stand for a
charmonium such as ${J/\psi,\eta_c,h_c,\chi_{cJ}}\cdots$ and a bottomonium such
as ${\Upsilon,\eta_b,h_b,\chi_{bJ}}\cdots$ respectively. If we call such a
collider with so high luminosity and running around the $Z$-boson resonance as
a $Z$-factory, then our results obtained here indicate that experimental
studies at a $Z$-factory about the various heavy quarkona (their ground and
excited states) via the two-body processes, especially, the production of the
bound states with quantum number $J^{PC}=1^{--}$ via ISR, have outstanding
advantages.
