  The generalized Weierstrass representation is used to analyze the asymptotic
behavior of a constant mean curvature surface that arises locally from an
ordinary differential equation with a regular singularity.
  We prove that a holomorphic perturbation of an ODE that represents a Delaunay
surface generates a constant mean curvature surface which has a properly
immersed end that is asymptotically Delaunay. Furthermore, that end is embedded
if the Delaunay surface is unduloidal.
