  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.

Differential Geometry