  We examine the equilibrium conditions of a curve in space when a local energy
penalty is associated with its extrinsic geometrical state characterized by its
curvature and torsion. To do this we tailor the theory of deformations to the
Frenet-Serret frame of the curve. The Euler-Lagrange equations describing
equilibrium are obtained; Noether's theorem is exploited to identify the
constants of integration of these equations as the Casimirs of the euclidean
group in three dimensions. While this system appears not to be integrable in
general, it {\it is} in various limits of interest. Let the energy density be
given as some function of the curvature and torsion, $f(\kappa,\tau)$. If $f$
is a linear function of either of its arguments but otherwise arbitrary, we
claim that the first integral associated with rotational invariance permits the
torsion $\tau$ to be expressed as the solution of an algebraic equation in
terms of the bending curvature, $\kappa$. The first integral associated with
translational invariance can then be cast as a quadrature for $\kappa$ or for
$\tau$.
