  We show that one of exact solutions of the second-order Dirac-Coulomb
equation were pseudo. In the corresponding original literature, it was
considered that the mass of the electron with a Coulomb potential was
position-dependent, but the obtained eigenvalues set was not the inevitable
mathematical deduction of the given second-order Dirac equation, and the
second-order Dirac equations were not the inevitable mathematical deduction of
the given couplet first-order Dirac equation with the position-dependent mass
of the electron. In the present paper, we obtain the correct solution of the
introduced first-order differential equations. This new solution would be
tenable only when the wave equation is correct, but there is not any experiment
date to validate the so-called position-dependent of the electron in the
Coulomb field.
