Can a Hamiltonian with non-spherical potential commute with l^2?

[Feelingfine]

I know that in the case of central potential V(r) the hamiltonian of the system always commutes with l^2 operator. But what happends in this case?

 

[A. Neumaier]

The Hamiltonian $H=L_z$ commutes with $L^2$ but is not rotation invariant. Adding a multiple of $H=L_z$ to a rotation invariant $H$ gives other counterexamples.

 

[hilbert2]

It's shown here that $\hat{L}^2$ and $\hat{x}^2$ commute

https://physics.stackexchange.com/questions/93533/commutator-of-l2-and-x2-p2

So put a particle in three dimensions to a field described by a harmonic oscillator potential, but only in the x-direction

$V(x,y,z) = ax^2$

That's not rotation invariant, but still commutes with the $\hat{L}^2$ (and hence the whole $\hat{H}$ commutes with $\hat{L}^2$).