This short review studies six two‑dimensional quantum systems in a flat plane that have more conserved quantities than usual. The authors analyze the Smorodinsky–Winternitz potentials I and II (the Holt potential), the Fokas–Lagerstrom model, the three‑body Calogero and Wolfes models (also called the A2 and G2 rational models), and the Tremblay–Turbiner–Winternitz (TTW) system for integer values of its index k. For each of these models they show the Schrödinger problem can be solved exactly by algebraic methods. This supports a conjecture, formulated in Montreal in 2001, that many maximally superintegrable systems in the plane are exactly solvable.

“Superintegrable” means a system has more independent conserved quantities than the number of coordinates. In quantum mechanics a conserved quantity is a differential operator that commutes with the Hamiltonian, the operator that represents the total energy. Having extra commuting operators constrains the motion and often makes the eigenvalue problem easier. “Exactly solvable” is used here in a specific, technical sense: the Hamiltonian preserves an infinite sequence of finite‑dimensional function spaces. On each space the eigenvalues and eigenfunctions can be found by algebraic means.

The authors show several concrete algebraic features that make these models solvable. For each model the Hamiltonian and its integrals can be written as differential operators with polynomial coefficients and without a constant term. The energy eigenfunctions can be taken as polynomials in variables built from the symmetry invariants of the discrete symmetry group of the system. Each model has a hidden Lie algebra, written g^(k) for various k, and the Hamiltonian and integrals lie in (or generate a subalgebra of) the universal enveloping algebra of that hidden algebra. The algebra of integrals is described as a four‑generator polynomial algebra with generators H, I1, I2 and I12 = [I1,I2]. The models also have infinitely many finite‑dimensional invariant subspaces that match finite representations of the hidden algebra.