An interesting class of axially symmetric surfaces, which generalizes Delaunay's unduloids and provides solutions of the shape equation is described in explicit parametric form. This class provide the first analytical examples of surfaces with periodic curvatures studied by K. Kenmotsu and leads to some unexpected relationships among Jacobian elliptic functions and their integrals.
Peter Djondjorov. Mariana Hadzhilazova. Ivaïlo Mladenov. Vassil M. Vassilev. "Beyond Delaunay Surfaces." J. Geom. Symmetry Phys. 18 1 - 11, 2010. https://doi.org/10.7546/jgsp-18-2010-1-11