  Exact solvability of some non-Hermitian $\eta$-weak-pseudo-Hermitian
Hamiltonians is explored as a byproduct of $\eta$-weak-pseudo-Hermiticity
generators. A class of V_{eff}(x)=V(x)+iW(x) potentials is considered, where
the imaginary part W(x) is used as an $\eta$-weak-pseudo-Hermiticity generator
to obtain exactly solvable $\eta$-weak-pseudo-Hermitian Hamiltonian models.
