On torsion-free groups with finite regular file bases

The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsion-free group $G$ has a finite system of generators $a_{1}$, ..., $a_{n}$ such that every element of $G$ has a unique presentation in the form $a_{1}^{k_{1}}\cdots a_{n}^{k_{n}}$ where $k_{i}\in \mathbb{Z}$, is it true that $G$ is virtually polycyclic? The answer is ``not always.'' A counterexample is constructed in this paper as a group presented by generators and defining relations.