Surfaces in the four-space and the Davey–Stewartson equations

We show that any equation from the Davey–Stewartson hierarchy induces an infinite family of geometrically different deformations of tori in R4 preserving the Willmore functional. We expose a derivation of the Weierstrass representation for surfaces in the four-space, which is not unique in difference from the case of surfaces in the three-space. This non-uniqueness implies that the spectral curve of a torus in R4 is not uniquely defined as a complex curve formed by the Floquet multipliers.