a module over $\cH(n)^{O(n)}$, which uses essentially the same ideas as [@LII].

First of all, there is an $O(n)$-invariant filtration on $\cH(n)$ such that the associated graded object $gr(\cH(n))$ is isomorphic to $S=Sym \bigoplus_{j\geq 0} V_j$ as a commutative ring, where $V_j\cong \mathbb{C}^n$ as $O(n)$-modules. In fact, $\cH(n)\cong S$ as vector spaces, and we view $\cH(n)$, equipped with the Wick product, as a deformation of $S$. Using Weyl’s first and second fundamental theorems of invariant theory for the standard representation of $O(n)$, we obtain a natural (infinite) strong generating set $\{\omega_{a,b}|~0\leq a\leq b\}$ for $\cH(n)^{O(n)}$, as well as an infinite set of relations among these generators. A linear change of variables produces a slightly more economical set of strong generators $\{j^{2m}|~m\geq 0\}$, where $j^{2m}$ has weight $2m+2$. In fact, $\cH(n)^{O(n)}$ is generated as a vertex algebra by $\{j^0, j^2\}$ for all $n\geq 1$, although this is only a strong generating set in the case $n=1$. The relation of minimal weight among $\{j^{2m}|~m\geq 0\}$ and their derivatives occurs at weight $n^2+3n+2$, and we conjecture that it gives rise to a decoupling relation $$j^{n^2+3n} = P(j^0,j^2,\dots,j^{n^2+3n-2}).$$ Here $P$ is a normally ordered polynomial in $j^0, j^2,\dots,j^{n^2+3n-2}$, and their derivatives. An easy consequence 