Commutative properties of singularly perturbate operators

Let a selfadjoint operator $A$ in Hilbert space $\mathcal H$ commutes with bounded operator $S$ and let $\widetilde A$ be singularly perturbate with respect to $A$, i.e. $\widetilde A$ coincides with $A$ on a dense domain in $\mathcal H$. The conditions under wich $\widetilde A$ commutes with $S$ are studied. The cases when $S$ is unbounded and when $S$ is replaced for singularly perturbate $\widetilde S$ are also investigated. As an example the Laplace operator in $L_2(\mathbf R^q)$ singularly perturbate by the set of $\delta$-functions and commuting with symmetrization in $\mathbf R^q$, $q=2,3$ or with regular representations of arbitrary isometric transformations in $\mathbf R^q$, $q\leqslant 3$ is considered.