Twisted conjugacy in $BS(n,1)$

In this article, we solve the twisted conjugacy problem for solvable Baumslag–Solitar groups $BS(n,1)$, i.e., we propose an algorithm which, given two elements $u,v\in BS(n,1)$ and an automorphism $\phi \in \text{Aut}(BS(n,1))$, decides whether $v=(w\phi {)}^{-1}uw$ for some $w\in BS(n,1)$. Also we prove that the automorphism group $\text{Aut}(BS(n,1))$ is orbit decidable – given two words on the generators $u,v\in F(X)$, decide whether the corresponding elements $u,v\in G$ can be mapped to each other by some automorphism in $\text{Aut}(BS(n,1))$.