Failure of finite isolated generation for an irreducible centralizer over a polycyclic group

Let \(G=\mathbb Z^2\rtimes_M\mathbb Z\), where \(M=\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)\), and let \(t\) generate the cyclic factor. The equation \([x,t]=1\) defines an infinite irreducible algebraic set over the two-generated torsion-free polycyclic group \(G\leq\mathrm{SL}_3(\mathbb Z)\). We prove that its radical in \(G*\langle x\rangle\) is not the normal isolated closure of any finite subset. Thus no finite equivalent system has radical equal to its normal isolated closure, giving a negative answer to Problem 14.22 of the Kourovka Notebook. The proof identifies the radical as the kernel of a scalar Laurent collection homomorphism. Alternating bilinear extensions then provide torsion-free separating groups for arbitrary finite families of radical equations.