CMC Surfaces with Branch Points
Branch points (where the derivative has rank 0)
are the only type of singularity for CMC surfaces in Euclidean 3-space. The first image is
the simplest type, with DPW potential off-diag$(z,1) \lambda^{-1} d z$.
The other singularity shown is a non-generic singularity of a spacelike CMC surface in Minkowki 3-space,
produced by the solution of the singular Björling problem
with Björling data $s(x)=x$, $t(x)=x$ and $\theta^\prime(x)=0.01$. It is like a branched swallowtail.