 Shifted two-dimensional oscillator in a uniform electric field

Completing the square in $H^{\prime }=\frac{1}{2}(p_x^2+p_y^2+x^2+y^2)-e\mathcal{E}x$ shifts the oscillator centre to $(a,0)$ and lowers every energy by $a^2/2$. Translated eigenfunctions keep their normalization. The angular momentum about the new centre is $L^{\prime }=(x-a)p_y-y{p}_x$, which commutes with $H^{\prime }$, whereas the angular momentum about the old centre has $[L,H^{\prime }]=-i\hslash ay$. For $a\ne 0$, a common normalizable eigenfunction of $L,H^{\prime }$ would have $y\psi =0$ and hence vanish in $L^2({\mathbb{R}}^2)$. This stronger argument excludes individual common eigenstates; noncommutativity alone generally excludes only a complete common eigenbasis.