Rationality of verbal subsets in solvable groups

A verbal subset of a group $G$ is a set $w[G]$ of all values of a group word $w$ in this group. We consider the question whether verbal subsets of solvable groups are rational in the sense of formal language theory. It is proved that every verbal subset $w[N]$ of a finitely generated nilpotent group $N$ with respect to a word w with positive exponent is rational. Also we point out examples of verbal subsets of finitely generated metabelian groups that are not rational.