Integrable and Superintegrable Extensions of the Rational Calogero-Moser Model in 3 Dimensions

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