$\mathbb{P}^3$. $\bar{D}\in |\mathcal{O}_{\bar{X}}(m)|$ and $\bar{\Lambda} \in |\mathcal{O}_{\bar{X}}(3-m)|$. $X$ is a double cover of $\mathbb{P}^2$ induced by $\Phi_{|K_{X}+\Lambda|}$. The branch curve is $2C+B$, where $C$,$B$ are reduced curves of degree $m$, $2$. $\Lambda\in |\Phi_{|K_{X}+\Lambda|}^*\mathcal{O}_{\mathbb{P}^2}(3-m)|$. $0<m \le 3$. ($p_g(X,\Lambda)=3$)

-   $X$ is the projection image of a Veronese embedding of $\mathbb{P}^2$. $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$ and $\bar{\Lambda}\in |\mathcal{O}_{\mathbb{P}^2}(4)|$. ($p_g(X,\Lambda)=5$)

-   $X$ is the projection image of $C_{N-1}\subset\mathbb{P}^N$, $N>3$. $\bar{D}$ is a sum of two lines passing the vertex, and $\bar{\Lambda}\in \mathcal{O}_{C_{N-1}}(2)$. ($p_g(X,\Lambda)=N$)

-   $X$ is the projection image of $\Sigma_d$ embedded in $\mathbb{P}^N$, $N>3$. $D\subset X$ is a line. We have 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 $\bar{\Lambda}\in |2\Delta_0+(2d+4)\Gamma|$. ($p_g(X,\Lambda)=N$)

Let $\bar{D}$ be a smooth quadric in $\mathbb{P}^2$ defined by $x^2+y^2+z^2=0$. $\tau$ acts on $\bar{D}$ by $x\mapsto -x, y \mapsto -y, z\mapsto z$. Let $\bar{\Lambda}=L_{x=0}+L_{y=0}$, where $L_{x=0}$, $L_{y=0}$ are two lines. Gluing $\mathbb{P}^2$ along $\bar{D}$ by $\tau$ we get a non-normal Gorenstein stable log surface $(X,\Lambda)$ with $(K_X+\Lambda)^2=p_g(X,\Lambda)-1=1$.

Let $\bar{D}$ be a smooth quartic in $\mathbb{P}^2$ defined by $x^4+y^4+z^4=0$. $\tau$ acts on $\bar{D}$ by $x\mapsto -x, y \mapsto -y, z\mapsto z$. Gluing $\mathbb{P}^2$ along $\bar{D}$ by $\tau$ we get a non-normal Gorenstein stable surface $X$ with $K_X^2=p_g-1=1$.

\[irrnonnormalstabledelta1\] Let 