  We use integrable systems techniques to study the singularities of timelike
non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski
3-space. The singularities arise at the boundary of the Birkhoff big cell of
the loop group involved. We examine the behaviour of the surfaces at the big
cell boundary, generalize the definition of CMC surfaces to include those with
finite, generic singularities, and show how to construct surfaces with
prescribed singularities by solving a singular geometric Cauchy problem. The
solution shows that the generic singularities of the generalized surfaces are
cuspidal edges, swallowtails and cuspidal cross caps.
