  Recently several new bounds of the lepton flavor violating decays of the
neutral vector mesons $J/\psi$ and $\Upsilon(nS)$ are reported from the
experiments. In the work, we analyze these processes in the scenario of the
minimal R-symmetric supersymmetric standard model by means of the effective
Lagrangian method. The predicted branching ratios are affected by the mass
insertion parameters $\delta^{ij}$ and the contributions from different parts
are comparable. Taking account of the constraints on the mass insertion
parameters from the experiments, the branching ratios for the most promising
processes $\Upsilon(nS)\rightarrow l\tau$ are predicted to be ten orders of
magnitude smaller than the present experimental bounds.
