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 ${\mathbb{R}}^4$ 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 ${\mathbb{R}}^4$ is not uniquely defined as a complex curve formed by the Floquet multipliers.