What are the possible magnetic fields with constant magnitude?

A now-deleted answer to this recent question prompted me to wonder about this and I can't find a clear answer in the top layer of google results, so I thought I'd ask here.

What are the possible magnetic fields with constant magnitude?

That is to say, suppose that $\mathbf B: \mathbb R^3 \to \mathbb R^3$ is

• solenoidal, so $\nabla \cdot \mathbf B = 0$, and


• with constant magnitude $|\mathbf B(\mathbf r)| \equiv B_0$.

What can be said about $\mathbf B$? Is the solenoidality condition strong enough to imply that $\mathbf B(\mathbf r)$ must be a constant vector field? Or is it possible for the direction of the vector field to change from point to point? If so, can a general description of this class of fields be formulated?