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On the elementary theory of an almost polycyclic group
Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 125-132 Cet article a éte moissonné depuis la source Math-Net.Ru

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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/

[1] M. I. Kargopolov, V. N. Remeslennikov, N. S. Romanovskii, V. A. Romankov, V. A. Churkin, “Algoritmicheskie problemy dlya $\mathcal O$-stepennykh grupp”, Algebra i logika, 8:6 (1969), 643–659 | MR

[2] Yu. L. Ershov, “Ob elementarnykh teoriyakh grupp”, DAN SSSR, 203:6 (1972), 1240–1243 | Zbl

[3] Yu. L. Ershov, “Nerazreshimost nekotorykh polei”, DAN SSSR, 161:1 (1965), 27–29 | Zbl

[4] R. Robinson, “Undecidable rings”, Trans. Amer. Math. Soc., 70:1 (1951), 137–159 | DOI | MR | Zbl

[5] J. Robinson, “The undecidability of algebraic rings and fields”, Proc. Amer. Math. Soc., 10:6 (1959), 950–957 | DOI | MR | Zbl

[6] Z. I. Borevich, I. R. Shafarevich, Teoriya chisel, izd-vo “Nauka”, Moskva, 1972 | MR