$\mathfrak{sl}(2)$-Trivial Deformations of $\operatorname{Vect}_{\mathrm{Pol}}(\mathbb R)$-Modules of Symbols

We consider the action of $\operatorname{Vect}_{\mathrm{Pol}}(\mathbb R)$ by Lie derivative on the spaces of symbols of differential operators. We study the deformations of this action that become trivial once restricted to $\mathfrak{sl}(2)$. Necessary and sufficient conditions for integrability of infinitesimal deformations are given.