  We prove that the Lie algebra $\mathcal{S}(\mathcal{P}(E,M))$ of symbols of
linear operators acting on smooth sections of a vector bundle $E\to M,$
characterizes it. To obtain this, we assume that
$\mathcal{S}(\mathcal{P}(E,M))$ is seen as ${\rm C}^\infty(M)-$module and that
the vector bundle is of rank $n>1.$ We improve this result for the Lie algebra
$\mathcal{S}^1(\mathcal{P}(E,M))$ of symbols of first-order linear operators.
We obtain a Lie algebraic characterization of vector bundles with
$\mathcal{S}^1(\mathcal{P}(E,M))$ without the hypothesis of being seen as a
${\rm C}^\infty(M)-$module.
