Some properties of the shape-invariant two-dimensional Scarf II model

We consider the two-dimensional Scarf II quantum model in the framework of supersymmetric quantum mechanics. We systematize results previously obtained for this integrable system and derive some new properties. In particular, we show that the model is exactly or quasi-exactly solvable in different regions of the system parameter values. We find a degeneracy of the spectrum for some specific parameter values. We calculate the action of symmetry operators of the fourth order in momenta for arbitrary wave functions obtained using the double shape invariance.