One part power series annihilator

Question

This following appear in infinite variable but for discussion let's take everything is finite if it makes the answer simpler. \begin{equation} \label{PartitionfuncKS} \mathcal{Z}^{KS}(x)=\sum_{\substack{g\geq 0, n\geq 1\\ 2g-2+n >0 }}\frac{h^{2g-2+n}}{n!}\sum_{(\alpha_1 ,\ldots , \alpha_n) \in I^{n}}F_{g,n}[\alpha_1 , \ldots \alpha_n ]x_{\alpha_{1}}\cdots x_{\alpha_{n}}\quad(*) \end{equation} The above equation is annihilated by \begin{equation} L_{i}:= \hbar\partial^{i}-\frac{1}{2} \sum_{cd}A^{icd}x_{c}x_{d}-\sum_{cd}\hbar B_{c}^{id}\partial^{c}x_{d}-\frac{1}{2} \sum_{cd}\hbar^{2}C_{cd}^{i}\partial^{c}\partial^{d}-\hbar D^{i}, \end{equation} where $\partial^i: = \frac{\partial}{\partial x_i}$, $A,B,C,D$ are constant that is $L_i$ partial differential equation. The following generating function \begin{equation} \label{onept} \mathcal{Z}_{onept}^{KS}(x)=\sum_{\substack{g\geq 0,\\ 2g-1 >0 }}h^{2g-1}\sum_{\alpha_1 \in I}F_{g,n}[\alpha_1 ]x_{\alpha_{1}}\quad (**) \end{equation} can be derived from (*) from (**). we introduce a parameter $s$ alongside $x$. Hence $$ \mathcal{Z}_{onept}^{KS}(x) = \frac{\partial}{\partial s}\mathcal{Z}^{KS}(sx)\Bigg\lfloor_{s=0} $$ My question is what is the corresponding operator that annihilate $\mathcal{Z}_{onept}^{KS}(x)$. How does it related to $L_i$? Is there any literature regarding this?