  Darboux transformation is a powerful tool for the construction of new
solvable models in quantum mechanics. In this article, we discuss its use in
the context of physical systems described by $4\times4$ Dirac Hamiltonians. The
general framework provides limited control over the resulting energy operator,
so that it can fail to have the required physical interpretation. We show that
this problem can be circumvented with the reducible Darboux transformation that
can preserve the required form of physical interactions by construction. To
demonstrate it explicitly, we focus on distortion scattering and spin-orbit
interaction of Dirac fermions in graphene. We use the reducible Darboux
transformation to construct exactly solvable models of these systems where
backscattering is absent, i.e. the models are reflectionless.
