  The Lie algebra $\mathcal{D}$ of regular differential operators on the circle
has a universal central extension $\hat{\mathcal{D}}$. The invariant subalgebra
$\hat{\mathcal{D}}^+$ under an involution preserving the principal gradation
was introduced by Kac, Wang, and Yan. The vacuum $\hat{\mathcal{D}}^+$-module
with central charge $c\in\mathbb{C}$, and its irreducible quotient
$\mathcal{V}_c$, possess vertex algebra structures, and $\mathcal{V}_c$ has a
nontrivial structure if and only if $c\in \frac{1}{2}\mathbb{Z}$. We show that
for each integer $n>0$, $\mathcal{V}_{n/2}$ and $\mathcal{V}_{-n}$ are
$\mathcal{W}$-algebras of types $\mathcal{W}(2,4,\dots,2n)$ and
$\mathcal{W}(2,4,\dots, 2n^2+4n)$, respectively. These results are formal
consequences of Weyl's first and second fundamental theorems of invariant
theory for the orthogonal group $\text{O}(n)$ and the symplectic group
$\text{Sp}(2n)$, respectively. Based on Sergeev's theorems on the invariant
theory of $\text{Osp}(1,2n)$ we conjecture that $\mathcal{V}_{-n + 1/2}$ is of
type $\mathcal{W}(2,4,\dots, 4n^2+8n+2)$, and we prove this for $n=1$. As an
application, we show that invariant subalgebras of $\beta\gamma$-systems and
free fermion algebras under arbitrary reductive group actions are strongly
finitely generated.
