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The first measurement of the cross section for incoherent photonuclearproduction of J/$\psi$ vector meson as a function of the Mandelstam $|t|$variable is presented. The measurement was carried out with the ALICE detectorat midrapidity, $|y|<0.8$, using ultra-peripheral collisions of Pb nuclei at acentre-of-mass energy per nucleon pair $\sqrt{s_{\mathrm{NN}}} = 5.02$ TeV.This rapidity interval corresponds to a Bjorken-$x$ range $(0.3$$-$$1.4)\times10^{-3}$. Cross sections are reported in five $|t|$ intervals in the range$0.04<|t|<1$~GeV$^2$ and compared to the predictions of different models.Models that ignore quantum fluctuations of the gluon density in the collidinghadron predict a $|t|$-dependence of the cross section much steeper than indata. The inclusion of such fluctuations in the same models provides a betterdescription of the data.

First measurement of the $|t|$-dependence of incoherent J/$\psi$ photonuclear production