Quasivariety Generated by Free Metabelian and 2-Nilpotent Groups

Let $qG$ be a quasivariety generated by a group $G$ and $\mathcal N$ be a non-Abelian quasivariety of groups with a finite lattice of subquasivarieties. Suppose $\mathcal N$ is contained in a quasivariety generated by the following two groups: a free $2$-nilpotent group $F_2(\mathcal N_2)$ of rank 2 and a free metabelian (i. e., with an Abelian commutant) group $F_2(\mathcal A^2)$ of rank 2. It is proved that either $\mathcal N=q F_2(\mathcal N_2)$ or $\mathcal N=q F_2(\mathcal A^2)$ in this instance.