Let M be a finitely generated metabelian group explicitly presented in a variety A2 of all metabelian groups. An algorithm is constructed which, for every endomorphism ϕ ∈ End(M) identical modulo an Abelian normal subgroup N containing the derived subgroup M and for any pair of elements u, v ∈ M, decides if an equation of the form (xϕ)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 ϕ ∈ End(M)