  The collective properties of final state hadrons produced in the high
statistics $_{44}^{96}$Ru+$_{44}^{96}$Ru and $_{40}^{96}$Zr+$_{40}^{96}$Zr
collisions at $\sqrt{s_\mathrm{NN}} = 200~\mathrm{GeV}$ are found to be
significantly different. Such differences were argued to be precise probes of
the difference in nucleon distribution in the isobar nuclei. We investigate the
$J/\psi$ production in the isobar collision via a relativistic transport
approach. By comparing the isobar systems according to equal centrality bin and
equal multiplicity bin, we find that the yield ratio of $J/\psi$ is sensitive
to the differences in both the number of binary collisions and the medium
evolution. Besides, the elliptic flow $v_2$ of $J/\psi$ is qualitatively
different from the light hadrons, and the ratio between Ru+Ru and Zr+Zr
collisions is sensitive to the medium evolution. The charmonium production
provides an independent probe to study the nucleon distribution in the isobar
system.
