Additional constraints on quasi-exactly solvable systems

We consider constraints on two-dimensional quantum mechanical systems in domains with boundaries. The constraints result from the Hermiticity requirement for the corresponding Hamiltonians. We construct new two-dimensional families of formally exactly solvable systems. Taking the mentioned constraints into account, we show that the systems are in fact quasi-exactly solvable at best. Nevertheless, in the context of pseudo-Hermitian Hamiltonians, some of the constructed families are exactly solvable.