generated via the conformal approach. The other contribution comes from the fact that the first order gauge transformation we just performed has second order pieces of the form, $$\begin{aligned}
\widetilde{h}^{(2)}_{\mu\nu} & = &
h^{(2)}_{\mu\nu} \nonumber \\
& &  
 -\widetilde{h}^{(1)}_{\mu
\nu},_{\rho} \xi_{(1)}^{\rho} - \widetilde{h}^{(1)}_{\mu \rho}
\xi_{(1)}^{\rho},_{\nu} - \widetilde{h}^{(1)}_{\rho \nu}
\xi_{(1)}^{\rho},_{\mu} \\
& & - 
{1\over2}g_{\mu \nu},_{\sigma},_{\lambda} \xi_{(1)}^{\sigma} \xi_{(1)}^{\lambda} 
- g_{\mu \lambda},_{\sigma} \xi_{(1)}^{\sigma}
\xi_{(1)}^{\lambda},_{\nu} - g_{\lambda \nu},_{\sigma}
\xi_{(1)}^{\sigma} \xi_{(1)}^{\lambda},_{\mu} - g_{\sigma \lambda}
\xi_{(1)}^{\sigma},_{\mu} \xi_{(1)}^{\lambda},_{\nu} \nonumber\end{aligned}$$

The contribution to the second order Zerilli function coming from the nonlinear terms in the gauge transformation can be found after a straightforward but tedious calculation. The relevant contributions to the second order $\ell=2$ metric at $t=0$, are, $$\begin{aligned}
{}^{(2)}K &=& {2 \over 3} {P^2 R^2 \over M^2 r^3} \nonumber\\
{}^{(2)}H_2 &=&  {2 \over 3} {P^2R^2 (R-M)^2 \over M^2 r^3 (r-2M)}
\nonumber\\
{}^{(2)}h_1 &=& {1 \over 3} {P^2 R^{5/2} (R-M) \over M^2 r^{3/2}
(r-2M)} \nonumber\\
{}^{(2)}G &=& {1 \over 3} {P^2 R^2 \over M^2 r^3}\end{aligned}$$ where $R = (\sqrt{r}+\sqrt{r-2M})^2/4$.

Another contribution to this initial data comes from the second order corrections to the conformal factor, calculated above. These contributions to the second order $\ell=2$ metric all vanish, except for, $$\begin{aligned}
{}^{(2)}H_2 & = & 4 P^2 \left(  