The variety of metabelian $\mathscr D$-groups

It is proved that an arbitrary metabelian $R$-group is imbeddable in a metabelian $\mathscr D$-group. Varieties consisting of metabelian $\mathscr D$-groups are described and a number of theorems on the structure of a free metabelian $\mathscr D$-group are obtained. A certain specially defined ring – an analog of the group ring – plays an essential role in the proofs.