Darboux transforms of a homogeneous torus

Darboux transforms of a homogeneous torus which are tori of revolution  but not CMC in the 3-sphere [Le 2009]

Compare to a Delaunay torus in the 3-sphere (torus of revolution in 3-sphere with constant mean curvature). Interesting coincidence?

And a 3-lobed torus of revolution (not CMC in the 3-sphere). 

More lobes.....

CMC surfaces

Delaunay surface and Delaunay Bubbletons

Wente tori

Minimal surfaces

Helicoid and Catenoid

Willmore tori

Willmore cylinder and Willmore tori, all  obtained by a Darboux transform of a (covering of) the Clifford torus, [G-L 2009].

Hamiltonian Stationary Lagrangian tori