Francesco Calogero, in a series of papers cited, studied a few
interesting classical nonlinear oscillators for which an
irrelevant parameter (typically, a multiplier c of the Lagrangian)
becomes immediately relevant during their attempted quantization.
Peter Leach pays a deeper attention to the Lie symmetries of these
"c-isochronous" problems and shows that and how this knowledge may
help us to solve the related (i.e., linear and properly
operator-ordered) time-dependent Schroedinger equations. In a
brief discussion the close connection of these models with the
Ermakov-Pinney and harmo0nic-oscillator problems is emphasized.

All the readers addressed by the story written by Peter Leach in
Italy (and, as I acknowledge hereby, re-told to me in RSA
recently) might find its continuation in the subsequent review
written by the same author (with a co-author) in Greece, the
reading of which I equally warmly recommended a week ago (see
extended abstract MR 2117167).

MR2117267
Leach, P. G. L. The solution of some quantum nonlinear oscillators
with the common symmetry group ${\rm SL}(2,R)$. J. Phys. A 38 (2005),
no. 7, 1543--1552.