Maxwell solution in differential forms

Maxwell equation’s in differential form: $dE = 0 and (ast d ast)E = p$ in static situation.

Where $E in Omega^1(U)$, $p in Omega^0(U)$, $ast$ is hodge star operator, $U= mathbb{R}^3$

what does $E$ looks like in open subset $U = mathbb{R}^3 backslash L$ where L is a line ($z$-axis)?

Edit: If U = $ mathbb{R}^3$, and dE = $0$ then E is exact since de Rham coholomly of U = {0} thus, E = $dphi$ where $phi in Omega^0(U)$. Note that I am looking for general solution, so I think there is no need for boundary condition. Similarily, If U =$ mathbb{R}^3 backslash L$, and dE = $0$ then E = $aw + dphi$ where $ a in mathbb{R}$ and w is closed 1-form (non-exact) and $dphi$ is the exact form as before. This is because de Rham coholmolgy of U (in this case) is isomorphoric to $ mathbb{R}$. I am trying to figure out w.