  The Vortex Filament Equation (VFE) is part of an integrable hierarchy of
filament equations. Several equations part of this hierarchy have been derived
to describe vortex filaments in various situations. Inspired by these results,
we develop a general framework for studying the existence and the linear
stability of closed solutions of the VFE hierarchy. The framework is based on
the correspondence between the VFE and the nonlinear Schr\"odinger (NLS)
hierarchies. Our results show that it is possible to establish a connection
between the AKNS Floquet spectrum and the stability properties of the solutions
of the filament equations. We apply our machinery to solutions of the filament
equation associated to the Hirota equation. We also discuss how our framework
applies to soliton solutions.
