A Kaluza–Klein reduction of super-integrable systems

Given a super-integrable system in n degrees of freedom, possessing an integral which is linear in momenta, we use the “Kaluza–Klein construction” in reverse to reduce to a lower dimensional super-integrable system. We give two examples of a reduction from 3 to 2 dimensions. The constant curvature metric (associated with the kinetic energy) is the same in both cases, but with two different super-integrable extensions. For these, we use different elements of the algebra of isometries of the kinetic energy to reduce to 2-dimensions. Remarkably, the isometries of the reduced space can be derived from those of the 3-dimensional space, even though it requires the use of quadratic expressions in momenta.