Suppose that $X$ is an $m\times n$ matrix, $A$ and $B$ are $n\times m$ matrices. How can you solve $$AXB=X^\top.$$
Is there an explicit formulation of $X$ in terms of $A$ and $B$ that makes the above equation true?
Are there explicit formulations of $A$ and $B$ in terms of $X$ that make the above equation true?
Provided that you know how to vectorize ("flatten") a matrix and what are Kronecker product and commutation matrix, the equation $AXB=X^T$ can be rewritten as $$(B^T\otimes A-K^{(m,n)})\operatorname{vec}(X)=0,$$ which is just an ordinary system of $mn$ linear equations in $mn$ unknowns, but only larger in size.