  We show that any equation from the Davey--Stewartson hierarchy induces an
infinite family of geometrically different deformations of tori in $\R^4$
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 $\R^4$ is not uniquely defined as a complex
curve formed by the Floquet multipliers.
