On symmetric partial differential operators

Let $$\sigma _{1}, \dots , \sigma _{k}$$ be the elementary symmetric functions of complex variables $$x_{1}, x_{k}$$ . We say that $$F \in {\mathbb {C}}[\sigma _{k}]$$ is a trace function if their exists $$f {C}}[z]$$ such $$F(\sigma _{k}) = \sum _{j=1}^{k} f(x_{j})$$ for all {C}}^{k}$$ give an explicit finite family second order differential operators in Weyl algebra $$W_{2}:= _{k}]\langle \frac{\partial }{\partial _{1}}, _{k}}\rangle $$ which generates left ideal $$W_{2}$$ partial killing functions. The proof uses theorem analogous to usual and corresponding map symbols. As application, we obtain each integer k holonomic system quotient by whose local solutions are linear combinations branches multivalued root universal equation degree k: $$z^{k} + _{h=1}^{k} (-1)^{h}\sigma _{h}z^{k-h} 0$$