The localized induction hierarchy and the Lund-Regge equation

An evolution equation of a curve is constructed by summing up the infinite sequence of commuting vector fields of the integrable hierarchy for the localized induction equation (LIE). It is shown to be equivalent to the Lund-Regge equation. The intrinsic equations governing the curvature and torsion are deduced in the form of integrodifferential evolution equations. A class of exact solutions which correspond to the permanent forms of a curve evolving by a steady rigid motion are presented. The analysis of the solutions reveals that, given the shape, there are two speeds of motion, one of which has no counterpart in the case of the LIE.