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A study is presented of two-dimensional superintegrable systems separating inCartesian coordinates and allowing an integral of motion that is a fourth orderpolynomial in the momenta. All quantum mechanical potentials that do notsatisfy any linear differential equation are found. They do however satisfynonlinear ODEs. We show that these equations always have the Painlev\'eproperty and integrate them in terms of known Painlev\'e transcendents orelliptic functions.

Fourth order Superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials