  The Hodge star mean curvature flow on a 3-dimensional Riemannian or
pseudo-Riemannian manifold is a natural nonlinear dispersive curve flow in
geometric analysis. A curve flow is integrable if the local differential
invariants of a solution to the curve flow evolve according to a soliton
equation. In this paper, we show that this flow on $\mathbb{S}^3$ and
$\mathbb{H}^3$ are integrable, and describe algebraically explicit solutions to
such curve flows. The Cauchy problem of the curve flows on $\mathbb{S}^3$ and
$\mathbb{H}^3$ and its B\"acklund transformations follow from this
construction.
