Lubrication theory is employed to examine surface-tension-dominated flows that arise during the application of thin coatings, using pressurized dies, to axially symmetric fibres. In all cases, it is assumed that the clearance between the die exit and the fibre is small compared with the fibre diameter. Previous analyses have been concerned with flows controlled by axial curvature for which the resulting solutions are unique. The present investigation examines stationary flows in which both the axial and the azimuthal curvatures are comparable. It is shown that this situation develops when the applicator volume flow is sufficiently large. Moreover, as the volume flow is increased, spatially oscillatory menisci can exist such that the solution is not always unique. These results are new, and calculations are presented that determine the maximum die clearance below which the solution remains unique. Within this regime, surface oscillations do not occur and there is a monotonic decay to the final uniform coating thickness.