  Considering the superior performances of the future e-p colliders, LHeC and
FCC-eh, we discuss the feasibility of detecting the extra neutral scalar
$h_{2}$ and the light gauge boson $Z^{}_{\mu\tau}$, which are predicted by the
${U(1)}_{L^{}_{\mu} - L^{}_{\tau}}$ model. Taking into account the experimental
constraints on the relevant free parameters, we consider all possible
production channels of $h_{2}$ and $Z^{}_{\mu\tau}$ at e-p colliders and
further investigate their observability through the optimal channels in the
case of the beam polarization P($e^{-}$)= -0.8. We find that the signal
significance above 5$\sigma$ of $h_{2}$ as well as $Z^{}_{\mu\tau}$ detecting
can be achieved via
$e^{-}p\to{e^{-}jh_{2}(\to{Z_{\mu\tau}Z_{\mu\tau}})}\to~e^{-}j+/\!\!\!\!{E}^{}_{T}$
process and a 5$\sigma$ sensitivity of $Z^{}_{\mu\tau}$ detecting can be gained
via
$e^{-}p\to{e^{-}jh_{1}(\to{Z^{}_{\mu\tau}Z^{}_{\mu\tau}})\to}~e^{-}j+/\!\!\!\!{E}^{}_{T}$
process at e-p colliders with appropriate parameter values and a designed
integrated luminosity. However, the signals of $h_{2}$ decays into pair of SM
particles are difficult to be detected.
