Let $A,B\in\mathbb{C}^{n\times 3n}$ be two matrices, and denote the Kronecker matrix product by $\otimes$. The matrix $$ M= \begin{bmatrix} A \otimes I_n \\\\ I_n \otimes B\end{bmatrix} $$ has size $2n^2 \times 3n^2$, therefore in the generic case its kernel has dimension $n^2$.

Is there a simple characterization of this kernel, or a way to compute it by exploiting this structure?

This problem comes from a numerical linear algebra problem (two-parameter eigenvalues problems). Do matrices with this structure appear in other fields, or do they ring a bell to you?

It looks appealing to look for vectors in this kernel in the form $v=a \otimes b \otimes c$, with $a,c\in\mathbb{C}^{n}$, $b\in\mathbb{C}^{3}$, since in this way they would be compatible with both Kronecker product structures. We normalize by restricting one entry of each of $a,b,c$ to be $1$, so we are left with $(n-1) + 2 + (n-1) = 2n$ unknowns. The condition $Mv=0$ is equivalent to $M(a\otimes b)=0$ and $N(b\otimes c)=0$, which are precisely $2n$ equations. So it looks like we should have several complex solutions in the generic case. Is there a viable way to compute them?