  The Kronig-Penney model, an exactly solvable one-dimensional model of crystal
in solid physics, shows how the allowed and forbidden bands are formed in
solids. In this paper, we study this model in the presence of both strong
spin-orbit coupling and the Zeeman field. We analytically obtain four
transcendental equations that represent an implicit relation between the energy
and the Bloch wavevector. Solving these four transcendental equations, we
obtain the spin-orbital bands exactly. In addition to the usual band gap opened
at the boundary of the Brillouin zone, a much larger spin-orbital band gap is
also opened at some special sites inside the Brillouin zone. The $x$-component
of the spin-polarization vector is an even function of the Bloch wavevector,
while the $z$-component of the spin-polarization vector is an odd function of
the Bloch wavevector. At the band edges, the optical transition rates between
adjacent bands are nonzero.
