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Considering the superior performances of the future e-p colliders, LHeC andFCC-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 experimentalconstraints on the relevant free parameters, we consider all possibleproduction channels of $h_{2}$ and $Z^{}_{\mu\tau}$ at e-p colliders andfurther investigate their observability through the optimal channels in thecase of the beam polarization P($e^{-}$)= -0.8. We find that the signalsignificance above 5$\sigma$ of $h_{2}$ as well as $Z^{}_{\mu\tau}$ detectingcan 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 gainedvia$e^{-}p\to{e^{-}jh_{1}(\to{Z^{}_{\mu\tau}Z^{}_{\mu\tau}})\to}~e^{-}j+/\!\!\!\!{E}^{}_{T}$process at e-p colliders with appropriate parameter values and a designedintegrated luminosity. However, the signals of $h_{2}$ decays into pair of SMparticles are difficult to be detected.

The signatures of the new particles $h_2$ and $Z_{\mu\tau}$ at e-p colliders in the $U(1)_{L_\mu-L_\tau}$ model