  We prove global existence of solutions to the initial value problem for a
third order dispersive flow into compact locally Hermitian symmetric spaces.
The equation we consider generalizes two-sphere-valued completely integrable
systems modelling the motion of vortex filament. Unlike one-dimensional
Schr\"odinger maps, our third order equation is not completely integrable under
the curvature condition on the target manifold in general. The idea of our
proof is to exploit two conservation laws and an energy which is not
necessarily preserved in time but does not blow up in finite time.
