Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials

We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schr\"odinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that their angular dependent part $S(\theta)$ does not satisfy any linear differential equation. In this case it satisfies a nonlinear ODE that has the Painlev\'e property and its solutions can be expressed in terms of the Painlev\'e transcendent $P_6$. We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case.