Periodic solutions of a many-rotator problem in the plane (Q2747529)

The main results of this paper apply to the dynamical system \(\ddot z_n=i \omega\dot z_n+ \sum^N_{m,l=1} \dot z_m\dot z_lf_{nml} (\underline z)\), where \(z_n(t)\) is complex, \(\underline z=(z_1,z_2, \dots,z_N)\), and \(f_{nml} (\underline z)\) is analytic and satisfies the scaling condition \(f_{nml}(\lambda \underline z)=\lambda^{-1} f_{nml}(\underline z)\). It is known that this equation is completely integrable and that all solutions are completely periodic in certain special cases. The authors prove that there is a wider class of problems in which solutions are also completely periodic.