  If a curve in R^3 is closed, then the curvature and the torsion are periodic
functions satisfying some additional constraints. We show that these
constraints can be naturally formulated in terms of the spectral problem for a
2x2 matrix differential operator. This operator arose in the theory of the
self-focusing Nonlinear Schrodinger Equation.
  A simple spectral characterization of Bloch varieties generating periodic
solutions of the Filament Equation is obtained. We show that the method of
isoperiodic deformations suggested earlier by the authors for constructing
periodic solutions of soliton equations can be naturally applied to the
Filament Equation.
