This is inspired in a problem from Griffths's Electromagnetism book (3th edition).
The problem asks us to construct a vector field which is both solenoidal and irrotational (div-free and curl-free) everywhere.
Perhaps the simplest solution are conservative fields with a harmonic potential. That is, suppose $F$ is conservative, then $\exists\phi:\mathbb{R}^3\to\mathbb{R}\ $ such that: \begin{equation} {F}=\nabla\phi\end{equation} Then, if $\phi$ is at least $\mathcal{C}^2$, it's trivial that $\nabla\times(\nabla\phi)=0$. The condition for the divergence implies that it's harmonic: \begin{equation}\nabla\cdot(\nabla\phi)=\nabla^2\phi=0\end{equation} The problem of course asks for a closed form. This potential yields the condition: \begin{equation}\phi(x,y,z)=y(x^2-z^2)\end{equation} And it's easy to see that it has a nice symmetry to it, since you can freely swap the variables and it will still have a zero laplacian. Taking the gradient extracts the answer.
My question here is, is there a more general form for such "symmetric" potentials? There's obviously a lot of scalar fields we can get, but is there a rigorous way of getting then?