Let $A$ be a vector field on $\mathbb{R}^3$. I am interested in finding solutions of
$$ \nabla^2 A \times {\rm curl} A = 0,\\ \quad {\rm div} A = 0. $$
Are there any exact solutions with nonzero $\nabla^2 A$ and ${\rm curl} A$? If there are, what are they?
These equations are a toy model for describing magnetic fields around astrophysical black holes.
Combining the two equations gives us
$$\operatorname{curl} A \times \operatorname{curl} \operatorname{curl} A = 0$$
Letting $B = \operatorname{curl} A$ for the moment, this leads us to solve
$$\operatorname{curl} B = \lambda B$$
This has many such solutions, but without further context or details (such as boundary conditions, spherical symmetry, behavior at infinity) we don't have any direction as to where to look. One such solution would be
$$B = (-\cos z, -\sin z, 0) \implies A = (\cos z, \sin z, 0)$$