The diophantine problem in some metabelian groups

In this paper we show that the Diophantine problem in solvable Baumslag-Solitar groups BS(1, k) and in wreath products A | Z, where A is a finitely generated abelian group and Z is an infinite cyclic group, is decidable, i.e., there is an algorithm that, given a finite system of equations with constants in such a group, decides whether or not the system has a solution in the group.