A nonseparable quantum superintegrable system in 2D real Euclidean space

In this paper, we derive a nonseparable quantum superintegrable system in 2D real Euclidean space. The Hamiltonian admits no second order integrals of motion but does admit one third and one fourth order integral. We also obtain a classical superintegrable system with the same properties. The quantum system differs from the classical one by corrections proportional to $\hbar^2.$