$BM{S}_3$-like algebras via the $Z_N$-graded $u(1{)}^2$ Kac-Moody algebra

The Sugawara construction provides a natural way to construct the Virasoro algebra from a current algebra. It was shown in Ref.~\cite{Ghazi:2025oin} that for the $u(1{)}^2$ Kac-Moody current algebra, there exist additional constructions that exhibit a ${\mathbb{Z}}_N$-graded structure. Indeed, the space of such constructions defines a non-compact algebraic variety whose dimension depends on $N$. In this paper, we consider the compactification of these algebraic varieties by adding points at infinity to the non-compact part, and show that these points correspond precisely to generalizations of $BM{S}_3$-like algebras. More explicitly, for a ${\mathbb{Z}}_2$ grading, the corresponding algebra coincides with the $BM{S}_3$ algebra, which takes the form $\mathrm{V}\mathrm{i}\mathrm{r}⋊F$, where $F$ is an infinite abelian ideal of the full algebra. For $N>2$, we show that there exist generalizations of the standard $BM{S}_3$ algebra of the form $\mathrm{V}\mathrm{i}\mathrm{r}⋊F$, where $F$ is a nonabelian ideal that forms a nilpotent algebra of depth $r<N$. We further demonstrate that the depth of the algebra is related to the order of the singularity of the algebraic variety at that point. We also show that the polynomials defining the algebraic varieties exhibit a factorization property into linear factors, which, if true, classifies all $BM{S}_3$-like algebras. Finally, we study the central extensions of these algebras, which are consistent with the general structure of algebras corresponding to primary fields of conformal weight $h=2$.