  In a recent work we have proven the existence of degenerate solutions to the
Dirac equation, corresponding to an infinite number of different
electromagnetic fields, providing also some examples regarding massless
particles. In the present article our results are extended significantly,
providing degenerate solutions to the Dirac equation for particles with
arbitrary mass, which, under certain conditions, could be interpreted as pairs
of particles (or antiparticles) moving in a potential barrier with energy equal
to the height of the barrier and spin opposite to each other. We calculate the
electromagnetic fields corresponding to these solutions, providing also some
examples regarding both spatially constant electromagnetic fields and
electromagnetic waves. Further, we discuss some potential applications of our
work, mainly regarding the control of the particles outside the potential
barrier, without affecting their state inside the barrier. Finally, we study
the effect of small perturbations to the degenerate solutions, showing that our
results are still valid, in an approximate sense, provided that the amplitude
of the electromagnetic fields corresponding to the exact degenerate solutions
is sufficiently small.
