The two dominant, linearly independent surface-wave modes in a circular cylinder, which differ only by an azimuthal rotation of ½π and have equal natural frequencies, are nonlinearly coupled, both directly and through secondary modes. The corresponding, weakly nonlinear free oscillations are described by a pair of slowly modulated sinusoids, the amplitudes and phases of which are governed by a four-dimensional Hamiltonian system that is integrable by virtue of conservation of energy and angular momentum. The resulting solutions are harmonic in a particular, slowly rotating reference frame. Harmonic oscillations in the laboratory reference frame are realized for three special sets of initial conditions and correspond to a standing wave with a fixed nodal diameter and to two azimuthally rotating waves with opposite senses of rotation. The finite-amplitude corrections to the natural frequencies of these harmonic oscillations are calculated as functions of the aspect ratio d/a (depth/radius). A small neighbourhood of d/a = 0.1523, in which the natural frequency of the dominant axisymmetric mode approximates twice that of the two dominant antisymmetric modes, is excluded. Weak, linear damping is incorporated through a transformation that renders the evolution equations for the damped system isomorphic to those for the undamped system.