The twisted conjugacy problem for endomorphisms of metabelian groups

Let $M$ be a finitely generated metabelian group explicitly presented in a variety $\mathcal A^2$ of all metabelian groups. An algorithm is constructed which, for every endomorphism $\varphi\in\operatorname{End}(M)$ identical modulo an Abelian normal subgroup $N$ containing the derived subgroup $M'$ and for any pair of elements $u,v\in M$, decides if an equation of the form $(x\varphi)u=vx$ has a solution in $M$. Thus, it is shown that the title problem under the assumptions made is algorithmically decidable. Moreover, the twisted conjugacy problem in any polycyclic metabelian group $M$ is decidable for an arbitrary endomorphism $\varphi\in\operatorname{End}(M)$.