  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.
