A note on common zeroes of Laplace--Beltrami eigenfunctions

Let $\text{De}u+\text{la}u=\text{De}v+\text{la}v=0$, where $\text{De}$ is the Laplace–Beltrami operator on a compact connected smooth manifold $M$ and $\text{la}>0$. If $H^1(M)=0$ then there exists $p\in M$ such that $u(p)=v(p)=0$. For homogeneous $M$, $H^1(M)\mathrm{\ne }0$ implies the existence of a pair $u,v$ as above that has no common zero.