  This paper is devoted to a study of relativistic eigenstates of Dirac
particles which are simultaneously bound by a static Coulomb potential and
added linear confining potentials. It has recently been shown that, despite the
addition of radially symmetric, linear confining potentials, some specific
bound-state energies surprisingly retain their exact Dirac--Coulomb values (in
the sense of an "exact symmetry"). This observation raises pertinent questions
as to the generality of the cancellation mechanism. A Foldy-Wouthuysen
transformation is used to find the relevant nonrelativistic physical degrees of
freedom, which include additional spin-orbit couplings induced by the linear
confining potentials. The matrix elements of the effective operators obtained
from the scalar, and time-like confining potentials mutually cancel for
specific ratios of the prefactors of the effective operators, which must be
tailored to the cancellation mechanism. The result of the Foldy-Wouthuysen
transformation is used to explicitly show that the cancellation is accidental
and restricted (for a given Hamiltonian) to only one reference state, rather
than traceable to a more general relationship among the obtained effective
low-energy operators. Furthermore, we show that the cancellation mechanism does
not affect anti-particle (negative-energy) states.
