  In this paper, we define a class of new geometric flows on a complete
Riemannian manifold. The new flow is related to the generalized (third order)
Landau-Lifishitz equation. On the other hand it could be thought of a special
case of the Schr\"odinger-Airy flow when the target manifold is a K\"ahler
manifold with constant holomorphic sectional curvature. We show the local
existence of the new flow on a complete Riemannian manifold with some
assumptions on Ricci tensor. Moreover, if the target manifolds are Einstein or
some certain type of locally symmetric spaces, we obtain the global results.
