by the involution $\theta$ sending $\alpha\mapsto -\alpha$. In [@DNI] it was shown that the invariant subalgebra $\cH^+$ under $\theta$ is a $\cW$-algebra of type $\cW(2,4)$, and in particular is strongly generated by the Virasoro element $L$ and an element $J$ of weight four. Using this result, the authors described the Zhu algebra of $\cH^+$, which is a commutative algebra on two generators, and they classified the irreducible modules of $\cH^+$. In [@DNII], Dong-Nagatomo considered a higher-rank analogue $\cH(n)^+$ of $\cH^+$. In this notation, $\cH(n)$ is the rank $n$ Heisenberg algebra, which is just the tensor product of $n$ copies of $\cH$, and $\cH(n)^+$ is the invariant subalgebra under the $-1$ involution. Unlike the rank $1$ case, the Zhu algebra of $\cH(n)^+$ is nonabelian, and it is difficult to describe it completely by generators and relations. However, the authors obtained enough information about it to classify the irreducible modules of $\cH(n)^+$. This result is important for understanding the structure and representation theory of vertex algebras of the form $V_L^+$. Here $V_L$ is the lattice vertex algebra attached to some lattice $L$ of rank $n$, and $V_L^+$ is the invariant subalgebra under the $-1$ involution.

In this paper, we study general invariant 