  This article surveys the Weierstrass representation of surfaces in the three-
and four-dimensional spaces, with an emphasis on its relation to the Willmore
functional. We also describe an application of this representation to
constructing a new type of solutions to the Davey-Stewartson II equation. They
have regular initial data, gain one-point singularities at certain moments of
time, and extend to smooth solutions for the remaining times.
