Degenerate integrable systems on the plane with a cubic integral of motion

We propose a classification of the known two-dimensional Hamiltonian systems of the natural form possessing an additional integral of motion that is cubic in the momenta. For degenerate systems of the Stäckel type, the additional cubic integral has the form of a “generalized angular momentum”. This allows constructing $n$-dimensional degenerate systems of the Stäckel type with additional cubic integrals of motion.