vertex algebra, and $G$ is reductive, $\cV^G$ will be strongly finitely generated under fairly general circumstances. Isolated examples of this phenomenon have been known for some years (see for example [@BFH][@FKRW][@EFH][@DNI][@KWY]), although the first general results of this kind were obtained in [@LII], in the case where $\cV$ is the $\beta\gamma$-system $\cS(V)$, $bc$-system $\cE(V)$, or $bc\beta\gamma$-system $\cE(V)\otimes \cS(V)$, associated to $V = \mathbb{C}^n$. The strong finite generation property is a subtle and essentially quantum" phenomenon, and is generally destroyed by passing to the classical limit before taking invariants. Often, $\cV$ admits a $G$-invariant filtration for which $gr(\cV)$ is a commutative algebra with a derivation (i.e., an abelian vertex algebra), and the classical limit $gr(\cV^G)$ is isomorphic to $(gr(\cV))^G$ as a commutative algebra. Unlike $\cV^G$, $gr(\cV^G)$ is generally not finitely generated as a vertex algebra, and a presentation will require both infinitely many generators and infinitely many relations.

One of the most basic examples of an invariant vertex algebra was studied by Dong-Nagatomo in [@DNI]. Let $\cH$ denote the Heisenberg vertex algebra, which is generated by a field $\alpha$ satisfying the operator product expansion (OPE) relation $\alpha(z) \alpha(w)\sim (z-w)^{-2}$. Clearly the automorphism group $Aut(\cH)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, and is generated 