We examine the dynamics of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating around this axis. An asymptotic equation is derived, which takes into account two orders of the so-called lubrication approximation. The equation admits a solution describing a steady-state distribution of film around the cylinder, and we examine the stability of this solution. It is demonstrated that the linearized problem admits infinitely many harmonic solutions, all of which have real frequencies (are neutrally stable). On the other hand, there are non-harmonic solutions that ‘explode’ (develop singularities) in a finite time. Systems having solutions of both types simultaneously have never been described in the literature. An example of this phenomenon is the main result of this paper.