Consider square matrix $A \in R^{n, n}$ and suppose we want to solve (like actually numerically solve)
$$ (A \oplus A) X = b $$
where $\oplus$ is the Kronecker sum $A \oplus B = A \otimes I_B + I_A \otimes B$.
In general, this system will be $n^2 \times n^2$ but it feels like there should be some structure to use here to avoid solving in that high dimensional space.
If there is some structure, it probably helps to write $X=Y \otimes Z$ and similar for $b$.