= Solution
The plane separates the exterior into two half-domains. Use an <odd Legendre extension for an equatorial Dirichlet plane> in each. On the upper hemisphere, extend $u+2u^2$ to the odd function $u+2u|u|$ on the whole sphere. On the lower hemisphere, the appropriate odd extension is $u-2u|u|$. Define, for odd $\ell$,
$$
b_\ell=2(2\ell+1)\int_0^1t^2P_\ell(t)\,dt.
$$
The <orthogonality> and parity of the <Legendre polynomials> give the solution in both half-domains as
$$
\boxed{\Psi(r,\theta)=\frac{a^2}{r^2}u+\operatorname{sgn}(u)\sum_{\ell=1,3,5,\ldots}b_\ell\left(\frac ar\right)^{\ell+1}P_\ell(u),\quad u=\cos\theta.}
$$
Set its value on $u=0$ to zero. The first coefficients are $b_1=3/2$ and $b_3=7/12$. All odd <Legendre polynomials> vanish at zero, so the plane condition holds. On the sphere the sum is $2u|u|$, and multiplication by $\operatorname{sgn}(u)$ recovers $2u^2$ on either side. Thus the original sphere data and decay at infinity both hold. A jump of the normal <derivative> at the sheet is permitted; the sheet can support induced <surface charge density>.
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