  In this paper, by using the $G_2$-structure on Im$(\mathbb O)\cong\mathbb
R^7$ from the octonions $\mathbb O$, the $G_2$-binormal motion of curves
$\gamma(t,s)$ in $\mathbb R^7$ associated to the almost complex structure on
$\mathbb S^6$ is studied. The motion is proved to be equivalent to
Schr\"odinger flows from $\mathbb R^1$ to $\mathbb S^6$, and also to a
nonlinear Schr\"odinger-type system in three unknown complex functions that
generalizes the famous correspondence between the binormal motion of curves in
$\mathbb{R}^3$ and the focusing nonlinear Schr\"odinger equation. Some related
geometric properties of the surface $\Sigma$ in Im$(\mathbb O)$ swept by
$\gamma(t,s)$ are determined.
