Superintegrable systems on 3 dimensional conformally flat spaces

We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional quadratic first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a universal reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux–Koenigs type, whose integrals are reductions of those of the 3 dimensional system.