  In this paper we discuss constraints on two-dimensional quantum-mechanical
systems living in domains with boundaries. The constrains result from the
requirement of hermicity of corresponding Hamiltonians. We construct new
two-dimensional families of formally exactly solvable systems and applying such
constraints show that in real the systems are quasi-exactly solvable at best.
Nevertheless in the context of pseudo-Hermitian Hamiltonians some of the
constructed families are exactly solvable.
