  Based on accurate calculations of the flux of equivalent photons of the
proton and heavy nuclei and the pQCD framework for the gluon distribution in
the proton and nuclei, we analyze the rapidity and momentum transfer
distributions of coherent $J/\psi$ photoproduction in ultraperipheral proton-Pb
collisions at the LHC. We demonstrate that unlike the case of proton-proton
UPCs marred by certain theoretical uncertainties and experimental limitations,
after a cut excluding the region of small momentum transfers, ultraperipheral
proton-Pb collisions offer a clean way to study the gluon distribution in the
proton down to $x \approx 10^{-5}$. Our analysis of the momentum transfer
distributions shows that an interplay of $J/\psi$ production by low-energy
photons on the nucleus and by high-energy photons on the proton in proton-Pb
UPCs can result in some excess of events at small $p_t$ in a definite region of
the rapidity y.
