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The wonderful compactification $X_m$ of a symmetric homogeneous space of typeAIII$(2,m)$ for each $m \geq 4$ is Fano, and its blowup $Y_m$ along the uniqueclosed orbit is Fano if $m \geq 5$ and Calabi-Yau if $m = 4$. Using acombinatorial criterion for K-stability of smooth Fano spherical varietiesobtained by Delcroix, we prove that $X_m$ admits a K\"ahler-Einstein metric foreach $m \geq 4$ and $Y_m$ admits a K\"ahler-Einstein metric if and only if $m =4,5$.

K\"ahler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII