vertex algebras $\cH(n)^G$, where $G$ is an arbitrary reductive group of automorphisms of $\cH(n)$. In the case where the action of $G$ extends to $V_L$, an understanding of $\cH(n)^G$ is a necessary first step in studying $V_L^G$. The [*full*]{} automorphism group of $\cH(n)$ preserving a natural conformal structure is the orthogonal group $O(n)$. We begin by studying $\cH(n)^{O(n)}$, which coincides with $\cH^+$ in the case $n=1$. For an arbitrary reductive group $G\subset O(n)$, $\cH(n)^G$ is completely reducible as a module over $\cH(n)^{O(n)}$, and this module structure is an essential ingredient in our description. Our approach in this paper is quite parallel to our earlier study of invariant subalgebras of the $\beta\gamma$-system $\cS(V)$. The automorphism group of $\cS(V)$ preserving its conformal structure is $GL_n$, and $\cS(V)^{GL_n}$ is isomorphic to $\cW_{1+\infty,-n}$ by a theorem of Kac-Radul [@KR]. In [@LI], we studied $\cW_{1+\infty,-n}$ via classical invariant theory, and we use a similar method in Section \[secortho\] of this paper to study $\cH(n)^{O(n)}$. There are many parallels between these two vertex algebras; for example, they both have abelian Zhu algebras, which implies that their irreducible, admissible modules are all highest-weight modules. This observation is crucial in our description in Section \[secgeneral\] of $\cH(n)^G$ as 