In an introductory Quantum Mechanics textbook, I found the following statement:
For two Hamiltonians $H$ and $H'$, non commuting with each other, but commuting with the same group of translations ${\cal{T}} (\vec{R})$ an eigenvector of $H$ can't be an eigenvector of $H'$.
But I don't see how $[H,H']\neq 0$ implies that $[H,H']$ cannot vanish for a specific eigenvector $\alpha$ of $H$, making it a shared eigenvector with $H'$.