  We make a comparative and comprehensive study on the charmonium exclusive
productions at the $e^+e^-$ collider with the collision energy either round the
$Z^0$-boson mass for a super $Z$ factory or equals to 10.6 GeV for the $B$
factories as Belle and BABAR. We study the total cross sections for the
charmonium production via the exclusive processes $e^+e^- \to \gamma^*/Z^0 \to
H_{1}+H_{2}$ and $e^+e^- \to \gamma^*/Z^0 \to H_{1} +\gamma$, where $H_{1}$ and
$H_{2}$ represent the dominant color-singlet $S$-wave and $P$-wave charmonium
states respectively. Total cross sections versus the $e^+e^-$ collision energy
$\sqrt{s}$, together with their uncertainties, are presented, which clearly
show the relative importance of these channels. At the $B$ factory, the
production channels via the virtual $\gamma^*$ propagator are dominant over the
channels via the $Z^0$ propagator by about four orders. While, at the super $Z$
factory, due to the $Z^0$-boson resonance effect, the $Z^0$ boson channels
shall provide sizable or even dominant contributions in comparison to the
channels via the $\gamma^*$ propagator. Sizable exclusive charmonium events can
be produced at the super $Z$ factory with high luminocity up to $10^{36}{\rm
cm}^{-2}{\rm s}^{-1}$, especially for the channel of $e^+e^- \to Z^0 \to H_{1}
+\gamma$, e.g. by taking $m_c=1.50\pm0.20$ GeV, we shall have
$(5.0^{+0.8}_{-0.6})\times10^4$ $J/\psi$, $(7.5^{+1.1}_{-0.9})\times10^3$
$\eta_c$, $(6.2^{+3.3}_{-1.9})\times10^3$ $h_{c}$,
$(3.1^{+1.7}_{-0.9})\times10^2$ $\chi_{c0}$, $(2.2^{+1.0}_{-0.4})\times10^3$
$\chi_{c1}$, and $(7.7^{+4.1}_{-2.4})\times10^2$ $\chi_{c2}$ events by one
operation year. Thus, in addition to the $B$ factories as BABAR and Belle, such
a super $Z$ factory shall provide another useful platform for studying the
heavy quarkonium properties and for testing QCD theories.
