Summary: Kaplan and Yardeni have found very simple exact limit cycle solutions in cyclically symmetric systems of \(N\) oscillator equations with linear coupling in zero order of a perturbation parameter and nonlinear coupling in first order. In contrast with such solutions in other nonlinear systems, each of these limit cycles is a normal mode of the unperturbed equations, with no change in frequency. The sources of this simple behaviour are studied here with the equations expressed in terms of the normal mode co-ordinates of the unperturbed system rather than in the original co-ordinates. It is found that the simplicity of the Kaplan- Yardeni solutions arises partly from an additional symmetry of the perturbation terms beyond the cyclic symmetry and partly from the specific choices of the perturbations. Extension of their systems to arbitrary \(N\) leads to the result that all such sets of equations have similar simple limit cycles. More general cyclically symmetric sets of equations are also discussed, with limit cycle solutions whose frequencies are shifted from the zero-order values by easily calculated amounts or with solutions which are linear combinations of zero-order normal modes. The results are exact, obtained without the use of small- perturbation theory.