We discuss the feasibility of detecting the gauge boson of the $U(1)_{L_{mu}-L_{tau}}$ symmetry, which possesses a mass in the range between MeV and GeV, at the Belle-II experiment. The kinetic mixing between the new gauge boson $Z$ and photon is forbidden at the tree level and is radiatively induced. The leptonic force mediated by such a light boson is motivated by the discrepancy in muon anomalous magnetic moment and also the gap in the energy spectrum of cosmic neutrino. Defining the process $e^{+} e^{-} rightarrow gamma Z rightarrow gamma u bar{ u}~(missing~energy)$ to be the signal, we estimate the numbers of the signal and the background events and show the parameter region to which the Belle-II experiment will be sensitive. The signal process in the $L_{mu}-L_{tau}$ model is enhanced with a light $Z$, which is a characteristic feature differing from the dark photon models with a constant kinetic mixing. We find that the Belle-II experiment with the design luminosity will be sensitive to the $Z$ with the mass $M_{Z} lesssim 1 $ GeV and the new gauge coupling $g_{Z} gtrsim 8cdot 10^{-4}$, which covers a half of the unconstrained parameter region that explains the discrepancy in muon anomalous magnetic moment. The possibilities to improve the significance of the detection are also discussed.
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