Irreducible Algebraic Sets in Metabelian Groups

We present the construction for a $u$-product $G_1\circ G_2$ of two $u$-groups $G_1$ and $G_2$, and prove that $G_1\circ G_2$ is also a $u$-group and that every $u$-group, which contains $G_1$ and $G_2$ as subgroups and is generated by these, is a homomorphic image of $G_1\circ G_2$. It is stated that if $G$ is a $u$-group then the coordinate group of an affine space $G^n$ is equal to $G \circ F_n$, where $F_n$ is a free metabelian group of rank $n$. Irreducible algebraic sets in $G$ are treated for the case where $G$ is a free metabelian group or wreath product of two free Abelian groups of finite ranks.