 Common eigenstate of two angular-momentum components

A common eigenstate of $J_x$ and $J_y$ must satisfy $J_z\psi =0$ by their commutator. Applying the other angular-momentum commutators then forces both remaining eigenvalues to vanish. Thus the state lies in a spin-zero sector. Noncommuting components do not possess a general common eigenbasis, but spin-zero states are an important exception to the claim that they can never share any eigenstate.