Solution
= Solution
Since the <electric field> is <irrotational>, introduce an <electrostatic potential> with $\mathbf E=-\nabla\varphi$. <Gauss's law> becomes the <Poisson equation>
$$
\boxed{(\partial_x^2+\partial_y^2)\varphi=-\rho/\epsilon_0,\qquad\mathbf E=-\nabla\varphi.}
$$
The curl equation is automatic because the <curl of a gradient> vanishes. Independence of $z$ makes $E_z$ constant, and its required decay forces it to be zero, so a $z$-independent potential suffices.