Generating pairs of solvable groups

Any two basis of a finite dimensional vector space are related by a finite sequence of moves, namely the elementary moves of the  Gauss elimination process. Nielsen transfomations are group-theoretic analogs of Gauss's moves. A group may however possess unrelated generating n-tuples. Considering the case n=2, I will explain how to compute a complete set of invariants for the induced Nielsen equivalence relation in a class of abelian-by-cyclic groups.