We study the rimming flow of a viscoelastic film on the inner surface of a horizontal rotating cylinder. Simple lubrication theory is applied assuming that the Reynolds number is small and the liquid film is thin. For the steady-state flow of the Upper-Convected Maxwell (UCM) fluid the mathematical model reduces to a first order nonlinear ODE for the film thickness. We show that the liquid viscoelasticity changes the flow structure. In particular, the singularity observed for viscous liquids within the same lubrication approximation can be eliminated due liquid elasticity. We performed a detailed numeric analysis of the model and revealed some criticality regimes which are specific only for viscoelastic liquids.