  The Weierstrass representation for spheres in $\R^3$ and, in particular,
effective construction of immersions from data of spectral theory origin is
discussed. These data are related to Dirac operators on a plane and on an
infinite cylinder and these operators are just representations of Dirac
operators acting in spinor bundles over the two-sphere which is naturally
obtained as a completion of a plane or of a cylinder. Spheres described in
terms of Dirac operators with one-dimensional potentials on a cylinder are
completely studied and, in particular, for them a lower estimate of the
Willmore functional in terms of the dimension of the kernel of the
corresponding Dirac operator on a two-sphere is obtained. It is conjectured
that this estimate is valid for all Dirac operators on spheres and some
reasonings for this conjecture are discussed. In Appendix a criterion
distinguishing Weierstrass representations, of universal coverings of compact
surfaces of higher genera, converted into immersions of compact surfaces is
given.
