**Second order superintegrable systems and algebraic varieties
**

**Abstract:**

A classical (or quantum) superintegrable system is an integrable
Hamiltonian system on an *n*-dimensional Riemannian space with potential
that admits *2n-1* functionally independent constants of the motion
polynomial in the momenta, the maximum number possible. If the constants
are quadratic the system is second order superintegrable. The
Kepler-Coulomb system is the best known example. Such systems have
remarkable properties: multi-integrability and multi-separability, an
algebra of higher order symmetries whose representation theory yields
spectral information about the Schrödinger operator, deep connections
with special functions and with Quasi-Exactly Solvable systems. I will
survey the structure results for systems in 2 and 3 dimensions. and show
that for real and complex Euclidean spaces the classification problem
reduces to finding points on an algebraic variety.

Joint work with *E.
Kalnins, J. Kress,* and *G. Pogosyan*.