  We show that the PT symmetric Hamiltonians (and their generalizations defined
in the text) may be all assigned the projected (so called Feshbach or
effective) nonlinear Hamiltonians which are "locally" Hermitian. This implies
that many (if not all) of the bound-state energies may be real in a broad
domain of Hermiticity-violating interactions. A complexification of a
Winternitz's super-integrable example in two dimensions is conjectured as an
illustration.
