  We analyze the energy spectrum of graphene in the presence of spin-orbit
coupling and a unidirectionally periodic Zeeman field, focusing on the
stability and location of Dirac points it may support. It is found that the
Dirac points at the $K$ and $K'$ points are generically moved to other
locations in the Brillouin zone, but that they remain present when the Zeeman
field $\vec{\Delta}(x)$ integrates to zero within a unit cell. A large variety
of locations for the Dirac points is shown to be possible: when $\vec\Delta
\parallel \hat{z}$ they are shifted from their original locations along the
direction perpendicular to the superlattice axis, while realizations of
$\vec\Delta(x)$ that rotate periodically move the Dirac points to locations
that can reflect the orbit of the rotating electron spin as it moves through a
unit cell. When a uniform Zeeman field is applied in addition to a periodic
$\vec\Delta \parallel \hat{z}$ integrating to zero, the system can be brought
into a metallic, Dirac semimetal, or insulating state, depending on the
direction of the uniform field. The latter is shown to be an anomalous quantum
Hall insulator.
