Suppose $\mathbb{K}$ is a field of characteristic $0$. Let $S_n=\mathbb{K}[[x_1,\dots,x_n]]$ be the ring of formal power series in $n$ variables, and $W_n=\mathbb{K}[x_1,\dots,x_n,\partial_1,\dots,\partial_n]$ be the $n$-dimensional Weyl algebra, acting on $S_n$ by multiplication and formal partial differentiation. An element $f \in S_n$ is called differentially integral if $M_f=\{ T(f)|T \in W_n \}$ is a finitely generated $P_n=\mathbb{K}[x_1,\dots,x_n]$-module.

Denote the set of differentially integral elements by $DZ_n$. An observation is that $P_n$ is strictly contained in $DZ_n$. The following questions are asked:

1. Does $DZ_n$ form a strict subring of $S_n$? It is known that $S_n \backslash DZ_n$ is non-empty as in this post.
2. Since $W_n$ acts on $S_n$ linearly, can we say that $DZ_n$ is to some extent 'graded' by $W_n$ via equation sets like $T_i(f)=0$ if $DZ_n$ is an actual ring? The equation grading restricted to $P_n$ is precisely the multi-degree.
3. Replace $S_n$ with $L_n=\mathbb{K}((x_1,\dots,x_n))$, the field of Laurent series with finite negative terms in $n$ variables. This time require $M_f$ to span a finite-dimensional vector space over $R_n=\mathbb{K}(x_1,\dots,x_n)$ and denote the new set of $f$ by $DQ_n$. If $DZ_n$ is an actual ring, does $DQ_n$ coincide with the field of fractions of $DZ_n$? This comes from the observation that $R_n$ is contained in $DQ_n$.