Deformation of \(\mathfrak{sl}\, (2)\) and \(\mathfrak{osp}\, (1|2)\)-modules of symbols

We consider the \(\mathfrak{sl}\, (2)\)-module structure on the spaces of symbols of differential operators acting on the spaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this structure and prove that any formal deformation is equivalent to its infinitesimal part. We study also the super analogue of this problem getting the same results.