  We study the geometric properties of Darboux transforms of constant mean
curvature (CMC) surfaces and use these transforms to obtain an
algebro-geometric representation of constant mean curvature tori. We find that
the space of all Darboux transforms of a CMC torus has a natural subset which
is an algebraic curve (called the spectral curve) and that all Darboux
transforms represented by points on the spectral curve are themselves CMC tori.
The spectral curve obtained using Darboux transforms is not bi-rational to, but
has the same normalisation as, the spectral curve obtained using a more
traditional integrable systems approach.
