The motion of an autonomous Hamiltonian system with two degrees of freedom, close to a system with a cyclic coordinate, is considered. It is assumed that the generating system admits of a steady rotation, the corresponding equilibrium position of the reduced system being stable in the linear approximation. It is also assumed that there is an internal resonance in the system: the ratio of the natural frequency of small oscillations of the reduced system to the frequency of variation of the cyclic coordinate is close to an integer. Non-linear oscillations of the complete system in the neighbourhood of this steady rotation are investigated. Periodic motions are constructed and their bifurcation and stability are examined. Methods of KAM theory are used to study quasi-periodic motions of the system. As an example, the problem of the motion of a nearly dynamically symmetrical heavy rigid body along an absolutely smooth horizontal plane is investigated in the case of internal resonance.