  Nonlinear Riccati and Ermakov equations are combined to pair the energy
spectrum of two different quantum systems via the Darboux method. One of the
systems is assumed Hermitian, exactly solvable, with discrete energies in its
spectrum. The other system is characterized by a complex-valued potential that
inherits all the energies of the former one, and includes an additional real
eigenvalue in its discrete spectrum. If such eigenvalue coincides with any
discrete energy (or it is located between two discrete energies) of the initial
system, its presence produces no singularities in the complex-valued potential.
Non-Hermitian systems with spectrum that includes all the energies of either
Morse or trigonometric Poeschl-Teller potentials are introduced as concrete
examples.
