PARTIALLY SUPERINTEGRABLE SYSTEMS AND NONLINEAR HARMONIC OSCILLATORS F. Calogero Department of Physics, University of Roma "La Sapienza", 00185 Roma, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Roma The notion will be illustrated of "partially superintegrable systems", namely dynamical systems -- for instance, classical many-body problems -- that feature an open (hence having nonvanishing measure) domain in their phase space such that all the trajectories originating from it are completely periodic with the same period ("isochronous"). And a mechanism that generate such systems -- entailing they are rather common -- will be illustrated, with several explicit examples. These results originate from work done in collaboration with J.-P. Fran¨oise and with M. Sommacal. Recent work with V. I. Inozemtsev will also be reported, that entails the existence of "nonlinear harmonic oscillators", namely assemblies of many nonlinearly coupled oscillators the time evolution of which is characterized by equations of motion (possibly rotation-invariant in S-dimensional space with arbitrary S, possibly also translation-invariant) of Newtonian type ("acceleration equal force", with one-body linear velocity-dependent forces and many-body cubic forces), that feature solutions which are either singular (and correspond to special initial data), or nonsingular (hence generic) and completely periodic with the same period ("isochronous"): tertium non datur.