  The Darboux-Koenigs metrics in 2D are an important class of conformally flat,
non-constant curvature metrics with a single Killing vector and a pair of
quadratic Killing tensors. In [arXiv:1804.06904] it was shown how to derive
these by using the conformal symmetries of the 2D Euclidean metric. In this
paper we consider the conformal symmetries of the 3D Euclidean metric and
similarly derive a large family of conformally flat metrics possessing between
1 and 3 Killing vectors (and therefore not constant curvature), together with a
number of quadratic Killing tensors. We refer to these as generalised
Darboux-Koenigs metrics. We thus construct multi-parameter families of
super-integrable systems in 3 degrees of freedom. Restricting the parameters
increases the isometry algebra, which enables us to fully determine the Poisson
algebra of first integrals. This larger algebra of isometries is then used to
reduce from 3 to 2 degrees of freedom, obtaining Darboux-Koenigs kinetic
energies with potential functions, which are specific cases of the known
super-integrable potentials.
