Center of some solvable groups with one defining relation

Necessary and sufficient conditions for the center of a metabelian group with one defining relation to be nontrivial are found. The center of such a group is described. The center of a group of the form $F/N'\langle g\rangle^F$ is studied under certain conditions. By means of a new technique, the recent result of A. F. Krasnikov and the author on the center of a group of the above form is sharpened.