We use the Dorfmeister-Pedit-Wu construction to use three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane. An appendix, by the third author, describes the DPW potentials that appear to give CMC punctured spheres with a k Delaunay ends (k-noids): the evidence is experiemental at present. These can have both unduloidal and nodoidal ends.