  The first measurement of the cross section for incoherent photonuclear
production of J/$\psi$ vector mesons as a function of the Mandelstam $|t|$
variable is presented. The measurement was carried out with the ALICE detector
at midrapidity, $|y|<0.8$, using ultra-peripheral collisions of Pb nuclei at a
centre-of-mass energy per nucleon pair of $\sqrt{s_{\mathrm{NN}}} = 5.02$ TeV.
This rapidity interval corresponds to a Bjorken-$x$ range $(0.3-1.4)\times
10^{-3}$. Cross sections are given in five $|t|$ intervals in the range
$0.04<|t|<1$ GeV$^2$ and compared to the predictions by different models.
Models that ignore quantum fluctuations of the gluon density in the colliding
hadron predict a $|t|$-dependence of the cross section much steeper than in
data. The inclusion of such fluctuations in the same models provides a better
description of the data.
