  A model with $L_{\mu} -L_{\tau}$ gauge symmetry is the least constrained
model as a resolution to the disagreement of the muon anomalous magnetic moment
between the theoretical predictions and the experimental results. In this
paper, we discuss how well the $L_{\mu} - L_{\tau}$ model can be constrained by
looking for decays of the charged kaon associated with a $L_{\mu}-L_{\tau}$
gauge boson. More concretely, we consider searches for single muon tracks from
the decays of stopped charged kaon as in the E949 experiment. In our
conservative estimation, we find that the favored parameter region for the muon
anomalous magnetic moment can be tested by using a $10$ times larger number of
the stopped charged kaons and about a $100$ times better photon rejection rate
than the E949 experiment.
