Given $\textbf{A}\in\mathbb{R}^{2 \times 2}$ and the form of Sylvester's equation, $\textbf{AX}=\textbf{XB}$, where $\textbf{X}\in\mathbb{C}^{2 \times 2}$ is unknown, is it possible to find $\textbf{B}$? The problem seems underdetermined, but the form of $\textbf{A}$ is so tempting,
$$\textbf{A}=\begin{bmatrix} 0 & 1 \\ a & 0 \end{bmatrix}$$.
I guess what I'm wondering is whether it is possible to write down a post-multiplication given the explicit form of a pre-multiplication. Fiddling with the algebra seems to make me doubt it, but nothing I've done seems definitive, only obstructed.