Residually finite varieties of groups were completely described, in [1], by Ol'shanskii. He proved that a group variety is residually finite if and only if it is generated by a finite group with abelian Sylow subgroups.

The next question is: “Which varieties are locally residually finite?” Hall [9] proved that all finitely generated abelian-by-nilpotent groups are residually finite. Hall formulated a conjecture that his result can be extended to the class of abelian-by-poly cyclic groups. Jategaonkar [2] proved that finitely generated abelian-by-polycyclic groups are residually finite.

The following result was obtained by Groves [8]. Let Tp be the variety generated in the variety ℬpA by all 2-generated groups belonging to ZA2, (p an odd prime), and let T2 be the variety generated in the variety A by all 2-generated groups belonging to ZA2A.

Theorem 1(Groves) If W is a variety of metanilpotent groups then the following conditions are equivalent.

1. W does not contain any Tp.

2. W is locally residually finite.

3. All finitely generated groups in W satisfy the maximal condition for normal subgroups.

In [4] it was proved that for odd primes p the variety Tp coincides with ZA2 ∩ ℬpA, and T2 was also described in the language of identities.

Conjecture 1The only minimal, non-locally residually finite, varieties of solvable groups are the varieties from the previous theorem and the varieties ApAqA (p, q are distinct primes).