Let $\mathbf F: \mathbb R^3 \rightarrow \mathbb R^3$ be a smooth, nonzero, divergence-free vector field. A simple computation shows that a sufficient condition for
$$ \operatorname{div} \left( \frac{\mathbf F}{|\mathbf F|} \right) = 0 $$
is that $\mathbf F \in \operatorname{Ker} J_{\mathbf F}$. Here $J_{\mathbf F}$ notes the Jacobian matrix of $\mathbf F$. Reformulating the former inclusion, we are led to consider the nonlinear system
$$ \begin{cases} (\mathbf F \cdot \nabla)\mathbf F= 0 \quad \text{in } \ \mathbb R^3 \\ \operatorname{div} \mathbf F = 0 \quad \text{in } \ \mathbb R^3, \end{cases} $$ that somehow resembles a degenerate/strange version of stationary Navier-Stokes (no Laplacian), which I am not quite familiar with. Are there any solutions (whatever this means) to this system? An alternative version of the problem woul be to work in a bounded domain $\Omega$ and add the natural boundary condition $\mathbf F \cdot \mathbf n = 0$ on $\partial \Omega$.