Algebraic sets in a finitely generated rigid $2$-step solvable pro-$p$-group

A $2$-step solvable pro-$p$-group $G$ is said to be rigid if it contains a normal series of the form
$$ G=G_1>G_2>G_3=1 $$
such that the factor group $A=G/G_2$ is torsion-free Abelian, and the subgroup $G_2$ is also Abelian and is torsion-free as a $\mathbb Z_pA$-module, where $\mathbb Z_pA$ is the group algebra of the group $A$ over the ring of $p$-adic integers. For instance, free metabelian pro-$p$-groups of rank $\ge2$ are rigid. We give a description of algebraic sets in an arbitrary finitely generated $2$-step solvable rigid pro-$p$-group $G$, i.e., sets defined by systems of equations in one variable with coefficients in $G$.