This is certainly a small step towards a satisfactory answer, but in the case of the solvable Baumslag-Solitar group $BS(1, n) = \langle a, b \, \vert \, aba^{-1} = b^n \rangle$ with $n \in \mathbb{Z} \setminus \{0\}$, anything can happen within its unique $T_2$-system. This [preprint][1] presents of proof of my claim, see Corollary E.


  [1]: http://arxiv.org/abs/1604.08896