The Lattice of Quasivarieties of Torsion-Free Metabelian Groups

Assume that a quasivariety $\mathcal M$ of groups contains a non-Abelian free metabelian group and a non-Abelian free nilpotent group of class 2. It is proved that the lattice of quasivarieties contained in $\mathcal M$ is infinite and non-modular.