following commutative diagram: $$\xymatrix{
 \bar{D}\ar@{^{(}->}[r] &\bar{X}\ar[d]^{\pi}\ar[drr]^{pr_c|_{\bar{X}}}\ar@{^{(}->}[rr]^{\Phi_{|K_{\bar{X}}+\bar{D}+\bar{\Lambda}|}} && \mathbb{P}^3 \ar@{-->}[d]^{pr_c}\\
  & X \ar[rr]_{\Phi_{|K_{X}+\Lambda|}} && \mathbb{P}^2
 }.$$ We see that $X$ is also a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branch curve ia $2C+B$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(3-m)|$.

If $pr_c|_{\bar{X}}$ is birational, it must be a normalisation map. Its image must be $X$ and $\pi=pr_c|_{\bar{X}}$.

Case VII) $\bar{X}$ is $\mathbb{P}^2$ embedding in $\mathbb{P}^5$. The non-normal locus $D$ is a line in $\mathbb{P}^5$ and $\bar{D}\in  |\mathcal{O}_{\mathbb{P}^2}(1)|$. $\bar{\Lambda}\in  |\mathcal{O}_{\mathbb{P}^2}(4)|$.

Case VIII) $\bar{X}$ is $C_{N-1}$, $N>3$. $\bar{D}$ is two rulings. $\bar{\Lambda}$ is linearly equivalent to $2H_{\infty}$ where $H_{\infty}$ is a hyperplane.

Case IX) $\bar{X}$ is $\Sigma_{d}$ embedded in $\mathbb{P}^N$ , $N>3$. Either $N=d+3$, $\bar{D}\in |\Delta_0+\Gamma|$ and $\bar{\Lambda}\in |2\Delta_0+(2d+2)\Gamma|$; or $N=d+5$, $\bar{D}=\Delta_0$ and $\bar{\Lambda}\in |2\Delta_0+(2d+4)\Gamma|$.

We summarize the above results in the following theorem:

\[delta1,0\] Let $(X,\Lambda)$ be an irreducible non-normal Gorenstein stable log surface as before. Assume $\Delta(X,K_X+\Lambda)=1$ and $\Delta(\bar{X},\pi^*(K_X+\Lambda))=0$. Then there are the following possibilities:

-   $\bar{X}$ is $\mathbb{P}^2$. $\bar{\Lambda}\in |\mathcal{O}_{\mathbb{P}^2}(2)|$ and $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(2)|$. Moreover, $c\not\in \bar{D}$. ($p_g(X,\Lambda)=2$)

-   $\bar{X}$ is $\mathbb{P}^2$. $\bar{\Lambda}=0$ and $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. Moreover, $c\not\in \bar{D}$. ($p_g(X,\Lambda)=2$)

-   $\bar{X}$ is $C_{N-1}$. $\bar{\Lambda}$ is two rulings and $\bar{D}\in \mathcal{O}_{C_{N-1}}(2)$. $c$ is the vertex of $C_{N-1}$. ($p_g(X,\Lambda)=N$)

-   $\bar{X}$ is a quadric in 