  We consider a model with gauged $L_\mu - L_\tau$ symmetry in which the
symmetry is spontaneously broken by a scalar field. The decay of the scalar
boson into two new gauge bosons is studied as a direct consequence of the
spontaneous symmetry breaking. Then, a possibility of searches for the gauge
and scalar bosons through such a decay at the LHC experiment is discussed. We
consider the case that the mass range of the gauge boson is $\mathcal{O}(10)$
GeV, which is motivated by anomalies reported by LHCb. We will show that the
signal significance of the searches for the gauge boson and scalar boson reach
to $3$ and $5$ for the scalar mixing $0.012$ and $0.015$, respectively.
