A classical (or quantum) superintegrable system on an n-dimensional
Riemannian manifold is an integrable Hamiltonian system with potential that
admits 2n-1 functionally independent constants of the motion that are
polynomial in the momenta, the maximum number possible. If these constants of
the motion are all quadratic, the system is second order superintegrable. Such
systems have remarkable properties. Typical properties are that 1) they are
integrable in multiple ways and comparison of ways of integration leads to new
facts about the systems, 2) they are multiseparable, 3) the second order
symmetries generate a closed quadratic algebra and in the quantum case the
representation theory of the quadratic algebra yields important facts about the
spectral resolution of the Schr\"odinger operator and the other symmetry
operators, and 4) there are deep connections with expansion formulas relating
classes of special functions and with the theory of Exact and Quasi-exactly
Solvable systems. For n=2 the author, E.G. Kalnins and J. Kress, have worked
out the structure of these systems and classified all of the possible spaces
and potentials. Here I discuss our recent work and announce new results for the
much more difficult case n=3.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA