Let $A \in \mathbb{C}^{n \times m}$ and $B \in \mathbb{C}^{m \times n}$ be matrices. Consider the following problem:
Find non-zero vectors $u \in \mathbb{C}^m$ and $v \in \mathbb{C}^n$ such that $Au = \lambda v$ and $Bv = \mu u$ for some $\lambda,\mu \in \mathbb{C}$.
Is there a name for this problem? Is it of interest in some area or application of mathematics?
Consider any $m \times n$ matrix $B$ and any $n \times n$ matrix $A$. Let $x$ be an eigenvector of $BA$, and let $y := Ax$. Then $Ax = y$ and $By = BAx = cx$ for some $c$, so $(x, y)$ is a solution of your problem. On the other hand, given a solution $(u, v)$ of your problem, we have that $x:= u$ is an eigenvector of $BA$, and $y := \lambda v$ is equal to $Ax$. So everything about this problem is already captured by the usual theory of eigenstuff.
(We could of course equally well start with $v$ and the matrix $AB$, instead.)