Geodesic
On the elementary theory of an almost polycyclic group
Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 125-132

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It is proved that the elementary theory of an almost polycyclic group is decidable if and only if this group is almost abelian. This generalizes the corresponding assertion on a finitely generated nilpotent group, proved earlier by Yu. L. Ershov. Bibliography: 6 titles. 
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N. S. Romanovskii. On the elementary theory of an almost polycyclic group. Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 125-132. http://geodesic.mathdoc.fr/item/SM_1981_39_1_a5/