  We make predictions for cross sections of $\rho$ and $\phi$ vector meson
photoproduction in ultraperipheral Xe-Xe collisions at $\sqrt{s_{NN}}=5.44$
TeV. Analyzing the momentum transfer distribution of $\rho$ mesons in this
process, we explore the feasibility of extracting the nuclear density of
$^{129}$Xe, which is needed in searches for dark matter with Xenon-based
detectors.
