  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.
