Identities of two-generated metabelian groups and finite conditions on Laurent ideals

Let $M$ be the free metabelian group of rank two and let $R=\\mathbb{Z}[X^{\\pm1},Y^{\\pm1}]$. We classify the varieties generated by two-generated metabelian groups by pairs $(n,I)$, where $n\\ge0$ and $I$ is an ideal of $R$. Admissibility is expressed by four power conditions and three families of Laurent substitutions. For each ideal generator $f$, the substitution conditions need only be checked on an integer box of side $6|\\mathrm{supp}(f)|$. Every admissible pair is realized by an explicit quotient of $M$, and two pairs give the same identities in all finite numbers of variables exactly when the pairs agree. This gives a finite-parameter classification for Kourovka Problem 8.54(b).