  We have studied self-conjugate second-order equations with spinor
wavefunctions for fermions moving in an external Coulomb field. For stationary
states, the equations are characterized by separated states with positive and
negative energies, which render probabilistic interpretation possible. For the
Coulomb field of attraction, the energy spectrum of the second-order equation
coincides with the spectrum of the Dirac equation, while the probability
densities of states are slightly different. For a Coulomb field of repulsion,
there exists an impermeable potential barrier with radius depending on the
classical electron radius and on the electron energy. The existence of the
impermeable barrier does not contradict the results of experiment for
determining the inner electron structure and does not affect (in the lowest
order of perturbation theory) the Coulomb electron scattering cross section.
The existence of the impermeable barrier can lead to positron confinement in
supercritical nuclei with $Z \geq 170$ in case of realization of spontaneous
emission of vacuum electron-positron pairs.
