  The 3-dimensional Heisenberg group can be equipped with three different types
of left-invariant Lorentzian metric, according to whether the center of the Lie
algebra is spacelike, timelike or null. Using the second of these types, we
study spacelike surfaces of mean curvature zero. These surfaces with
singularities are associated with harmonic maps into the 2-sphere. We show that
the generic singularities are cuspidal edge, swallowtail and cuspidal
cross-cap. We also give the loop group construction for these surfaces, and the
criteria on the loop group potentials for the different generic singularities.
Lastly, we solve the Cauchy problem for harmonic maps into the 2-sphere using
loop groups, and use this to give a geometric characterization of the
singularities. We use these results to prove that a regular spacelike maximal
disc with null oundary must have at least two cuspidal cross-cap singularities
on the boundary.
