Nonlinear harmonic oscillators

The existence is noted of assemblies of an arbitrary number of complex oscillators, or equivalently, of an arbitrary even number of real oscillators, characterized by Newtonian equations of motion ('acceleration equal force') with one-body velocity-dependent linear forces and many-body velocity-independent cubic forces, all the nonsingular solutions of which are isochronous (completely periodic with the same period). As for the singular solutions, as usual they emerge, in the context of the initial-value problem, from a closed domain in phase space having lower dimensionality