Integrability of a globally coupled oscillator array

We show that a dynamical system of N phase oscillators with global cosine coupling is completely integrable. In particular, we prove that the N-dimensional phase space is foliated by invariant two-dimensional tori, for all N≥3. Explicit expressions are given for the constants of motion, and for the solitary waves that occur in the continuum limit. Our analysis elucidates the origin of the remarkable phase space structure detected in recent numerical studies of globally coupled arrays of Josephson junctions, lasers, and Ginzburg-Landau oscillators.