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).
Delaunay surface and Delaunay Bubbletons
Helicoid and Catenoid
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
Castro Urbano tori [L-R 2008]