We analyse the behaviour of the Dirac equation in $d=1+1$ with Lorentz scalar
potential. As the system is known to provide a physical realization of
supersymmetric quantum mechanics, we take advantage of the factorization method
in order to enlarge the restricted class of solvable problems. To be precise,
it suffices to integrate a Ricatti equation to construct one-parameter families
of solvable potentials. To illustrate the procedure in a simple but relevant
context, we resort to a model which has proved useful in showing the phenomenon
of fermion number fractionalization.