Log-canonical pairs and Gorenstein stable surfaces with $K_X^2=1$

We classify log-canonical pairs $(X,\mathrm{\Delta })$ of dimension two with $K_X+\mathrm{\Delta }$ an ample Cartier divisor with $(K_X+\mathrm{\Delta }{)}^2=1$, giving some applications to stable surfaces with $K^2=1$. A rough classification is also given in the case $\mathrm{\Delta }=0$.