= Solution
Let $Y=\begin{pmatrix}p&q\\r&s\end{pmatrix}$. The <commutativity> condition gives
$$
XY=\begin{pmatrix}2r&2s\\-p&-q\end{pmatrix},\qquad
YX=\begin{pmatrix}-q&2p\\-s&2r\end{pmatrix}.
$$
Equality is equivalent to $s=p$ and $q=-2r$. Therefore
$$
\boxed{Y=\begin{pmatrix}\alpha&2\beta\\-\beta&\alpha\end{pmatrix}
=\alpha I+\beta X,\qquad \alpha,\beta\in\mathbb R.}
$$
Each displayed <matrix> commutes with $X$, since it is a polynomial in $X$. The <centralizer of a nonscalar two by two matrix> explains why this two-parameter form is typical.
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