  The problem of a fermion subject to a general mixing of vector and scalar
potentials in a two-dimensional world is mapped into a Sturm-Liouville problem.
Isolated bounded solutions are also searched. For the specific case of an
inversely linear potential, which gives rise to an effective Kratzer potential
in the Sturm-Liouville problem, exact bounded solutions are found in closed
form. The case of a pure scalar potential with their isolated zero-energy
solutions, already analyzed in a previous work, is obtained as a particular
case. The behaviour of the upper and lower components of the Dirac spinor is
discussed in detail and some unusual results are revealed. The nonrelativistic
limit of our results adds a new support to the conclusion that even-parity
solutions to the nonrelativistic one-dimensional hydrogen atom do not exist.
