Two problems for solvable and nilpotent groups

Section 1 gives a brief review of known results on embeddings of solvable, nilpotent, and polycyclic groups in $2$-generated groups from these classes, including the description of the author's recently obtained solution to the Mikaelian–Ol'schanskii problem on embeddings of finitely generated solvable groups of derived length $l$ in solvable groups of derived length $l+1$ with a fixed small number of generators.
Section 2 contains a somewhat more extensive review of known results on the rational subset membership problem for groups, including the presentation of the author's recently obtained solution to the Laurie–Steinberg–Kambites–Silva–Zetsche problem of whether the membership problem is decidable for finitely generated submonoids of free nilpotent groups.