I know that the Clifford torus in $S^3$ is Willmore. I can also think about $S^1$-equivariant tori and Hopf tori as Willmore in the $3$-sphere.


Does anyone know if there are more Willmore tori in $S^3$? Any source, idea or comment is very welcome.



Willmore tori in $S^3$
Yes, there are many examples . First of all, there are infinitely many minimal tori (immersed, not embedded ) in the 3-sphere.
There are also minimal surfaces with embedded planar ends of genus one.
Also, there is the Babich-Bobenko paper in which Willmore tori with umbilical lines are constructed.


