On a closed manifold $M$, let $P(f)=0$ be a linear overdetermined elliptic system of PDE of 2nd order with smooth coefficients. By overdetermined ellipticity, I mean the principal sympbol is injective. For simplicity, let's assume it is for a real-valued function $f$. If a solution $f$ is zero on an open subset, then is $f$ zero on the whole of $M$? If this is not true in general, what would be conditions for it to be true? Also please recommend some references. Thanks a lot!