  In this work, we calculate the total cross sections and the center-of-mass
frame angular distributions of the charmonium production plus a light meson by
the low energy $p\bar{p}$ interaction. The results of $p\bar{p}\to \pi^0 \Psi$
with and without form factor (FF) indicate that the FF contribution in the
calculation cannot be ignored. The obtained cross section of $p\bar{p}\to \pi^0
J/\psi$ with FF can fit the E760 data well. We also predict the total cross
sections and the center-of-mass frame angular distributions of $p\bar{p}\to
\omega \Psi$, which show that these physical quantities are dependent on Pauli
($g_\omega$) and Dirac ($\kappa_\omega$) coupling constants of the $pp\omega$
interaction. Thus, $p\bar{p}\to \omega \Psi$ can be as the ideal channel to
test the different theoretical values of $g_\omega$ and $\kappa_\omega$.
Applying the formulae of $p\bar{p}\to \pi^0 \Psi$ and $p\bar{p}\to \omega
\Psi$, we predict the total cross sections of the $p\bar{p}\to \eta \Psi$ and
$p\bar{p}\to \rho \Psi$ reactions. Our results show a common behavior of the
charmonium production with a light meson by the $p\bar{p}$ interaction, where
the total cross section of the $\eta_c$ production is the largest one among all
discussed processes. The above observations can be directly tested at the
forthcoming $\bar{P}$ANDA experiment.
