  A study is presented of two-dimensional superintegrable systems separating in
Cartesian coordinates and allowing an integral of motion that is a fourth order
polynomial in the momenta. All quantum mechanical potentials that do not
satisfy any linear differential equation are found. They do however satisfy
nonlinear ODEs. We show that these equations always have the Painlev\'e
property and integrate them in terms of known Painlev\'e transcendents or
elliptic functions.
