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Calculations and predictions are presented within the framework of the statistical hadronization model for transverse momentum spectra of the charmonium states J/ψ, ψ(2S) and X(3872) produced in nucleus-nucleus collisions at LHC energies. The results are confronted with available data and exhibit very good agreement by using particle flow profiles from state-of-the-art hydrodynamic calculations. For X(3872) production in Pb–Pb collisions we predict a transverse momentum distribution similar in shape to that for J/ψ with a strong enhancement at low transverse momenta and a production yield of about 1% relative to that for J/ψ. Keywords: Relativistic nuclear collisions, Statistical hadronization, Charmonium