A freedom theorem for groups with one defining relation in the varieties of solvable and nilpotent groups of given lengths

The freedom theorem of Magnus is well known: if a group $G$ is given by generators $x_1,x_2,\dots$ and a single defining relation $r=1$, and if $r$ is not conjugate to any word in $x_2,\dots$, then the elements $x_2,\dots$ freely generate in $G$ a free subgroup. In this note analogous theorems of Magnus are established for groups given by one defining relation in the varieties of soluble and nilpotent groups of given lengths.
Bibliography: 7 titles.