Integrable and superintegrable extensions of the rational Calogero-Moser model in 3 dimensions

We consider a class of Hamiltonian systems in 3 degrees freedom, with particular type quadratic integral and which includes the rational Calogero-Moser system as case. For general class, we introduce separation coordinates to find separable (and therefore Liouville integrable) system, two integrals. This gives coupling large potentials, generalising series potentials are parabolic coordinates. Particular cases {\em superintegrable}, including Kepler resonant oscillator. The initial calculations paper concerned flat (Cartesian type) kinetic energy, but Section \ref{sec:conflat-general}, conformal factor} $\varphi$ $H$ extend integrals this All previous results generalised then some 2 dimensional symmetry algebras Kinetic energy (Killing vectors), restrict factor. enables us reduce our from giving rise many interesting systems, both H\'enon-Heiles on Darboux-Koenigs $D_2$ background.