The periodic s1-equivariant hypersurfaces of constant mean curvature can be obtained by using the Lagrangians with suitable potential functions in the Berger spheres. In the corresponding Hamiltonian system, the conservation law is effectively applied to the construction of periodic s1-equivariant surfaces of arbitrary positive constant mean curvature.
 W-Y. Hsiang, “On Generalization of Theorems of A. D. Alexandrov and C. Delaunay on Hypersurfaces of Constant Mean Curvature,” Duke Mathematical Journal, Vol. 49, No. 3, 1982, pp. 485-496. doi:10.1215/S0012-7094-82-04927-4
 H. Muto, Y. Ohnita and H. Urakawa, “Homogeneous Minimal Hypersurfaces in the Unit Spheres and the First Eigenvalues of Their Laplacian,” Tohoku Mathematical Journal, Vol. 36, No. 2, 1984, pp. 253-267. doi:10.2748/tmj/1178228851