Finitely presented and solvable not implies polycyclic

Statement
It is possible to have a finitely presented solvable group $$G$$ -- a group $$G$$ that is both a finitely presented group and a solvable group -- such that $$G$$ is not a polycyclic group. In particular, this means that $$G$$ is not a Noetherian group, i.e., it has a subgroup that is not finitely generated.

Related facts

 * Finitely generated and solvable not implies finitely presented

Proof
The group $$G = BS(1,2)$$, the Baumslag-Solitar group with parameters $$(1,2)$$, works:

$$\langle a,b \mid bab^{-1} = a^2 \rangle$$

The normal closure of $$\langle a \rangle$$ in this group, which is also the derived subgroup of $$G$$, is isomorphic to the group of 2-adic rationals, i.e., the group of all rationals with denominators powers of 2.