= Solution
The planar <orbital angular momentum> is
$$
L=xp_y-yp_x
=i(a_y^\dagger a_x-a_x^\dagger a_y).
$$
The canonical commutation relations give
$$
[H,x]=-ip_x,
\quad [H,p_x]=ix,
\quad [H,y]=-ip_y,
\quad [H,p_y]=iy.
$$
Consequently
$$
\begin{aligned}
[H,L]
&=[H,x]p_y+x[H,p_y]-[H,y]p_x-y[H,p_x]\\
&=-ip_xp_y+ixy+ip_yp_x-iyx=0.
\end{aligned}
$$
Thus angular momentum preserves each energy eigenspace. More explicitly,
$$
0=\langle E|[H,L]|E'\rangle
=(E-E')\langle E|L|E'\rangle,
$$
so
$$
\boxed{\langle E|L|E'\rangle=0\quad\hbox{if }E\ne E'.}
$$
Within a degenerate energy eigenspace, this argument places no further restriction on the matrix of $L$.
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