Recursion operators, higher-order symmetries and superintegrability in quantum mechanics

A connection between the theory of superintegrable quantum-mechanical systems, which admit a maximal number of integrals of motion, and the standard Lie group theory is established. It is shown that the flows generated by first- and second-order Lie symmetries of the bidimensional Schrodinger equation can be classified and interpreted as quantum-mechanical operators which commute with integrable or superintegrable Hamiltonians. In this way! ail known superintegrable potentials in the plane are naturally obtained and slightly more general integrals of motion are found.