  We consider an $U(1)_{L_\mu -L_\tau}$ extended left-right symmetric gauge
theory where the neutrino masses are generated through inverse seesaw
mechanism. In this model the muon $(g-2)$ anomaly is accounted for by the
mediation of $Z_{\mu\tau}$, the gauge boson of $U(1)_{L_\mu - L_\tau}$
symmetry. The symmetries of the model require the light neutrino mass matrix to
have a particular two-zero texture, which leads to non-trivial constraints in
the minimum neutrino mass. In addition, the model predicts observable charged
lepton flavor violation in $\mu-\tau$ sector.
