  A Hamiltonian with two degrees of freedom is said to be superintegrable if it
admits three functionally independent integrals of the motion. This property
has been extensively studied in the case of two-dimensional spaces of constant
(possibly zero) curvature when all the independent integrals are either
quadratic or linear in the canonical momenta. In this article the first steps
are taken to solve the problem of superintegrability of this type on an
arbitrary curved manifold in two dimensions. This is done by examining in
detail one of the spaces of revolution found by G. Koenigs. We determine that
there are essentially three distinct potentials which when added to the free
Hamiltonian of this space have this type of superintegrability. Separation of
variables for the associated Hamilton-Jacobi and Schroedinger equations is
discussed. The classical and quantum quadratic algebras associated with each of
these potentials are determined.
