  Darboux transformations are employed in construction and analysis of Dirac
Hamiltonians with pseudoscalar potentials. By this method, we build a four
parameter class of reflectionless systems. Their potentials correspond to
composition of complex kinks, also known as twisted kinks, that play an
important role in the $1+1$ Gross-Neveu and Nambu-Jona-Lasinio field theories.
The twisted kinks turn out to be multi-solitonic solutions of the integrable
AKNS hierarchy. Consequently, all the spectral properties of the Dirac
reflectionless systems are reflected in a non-trivial conserved quantity, which
can be expressed in a simple way in terms of Darboux transformations. We show
that the four parameter pseudoscalar systems reduce to well-known models for
specific choices of the parameters. An associated class of transparent
non-relativistic models described by matrix Schr\"odinger Hamiltonian is
studied and the rich algebraic structure of their integrals of motion is
discussed.
