Partially divisible completions of rigid metabelian pro-$p$-groups

Previously, the author defined the concept of a rigid (abstract) group. By analogy, a metabelian pro-$p$-group $G$ is said to be rigid if it contains a normal series of the form $G=G_1\ge G_2\ge G_3=1$ such that the factor group $A=G/G_2$ is torsion-free Abelian, and $G_2$ being a $\mathbb Z_pA$-module is torsion-free. An abstract rigid group can be completed and made divisible. Here we do something similar for finitely generated rigid metabelian pro-$p$-groups. In so doing, we need to exit the class of pro-$p$-groups, since even the completion of a torsion-free nontrivial Abelian pro-$p$-group is not a pro-$p$-group. In order to not complicate the situation, we do not complete a first factor, i.e., the group $A$. Indeed, $A$ is simply structured: it is isomorphic to a direct sum of copies of $\mathbb Z_p$. A second factor, i.e., the group $G_2$, is completed to a vector space over a field of fractions of a ring $\mathbb Z_pA$, in which case the field and the space are endowed with suitable topologies. The main result is giving a description of coordinate groups of irreducible algebraic sets over such a partially divisible topological group.