Embedding of Baumslag–Solitar groups into the generalized Baumslag–Solitar groups

A finitely generated group $G$ that acts on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group or GBS-group. Let $p$ and $q$ be coprime integers other than $0,1$, and $-1$. We prove that the Baumslag–Solitar group $BS(p,q)$ embeds into $G$ if and only if the equation $x^{-1}y^px=y^q$ is solvable in $G$ for $y\ne1$ i.e., $\frac pq\in\Delta(G)$, where $\Delta$ is the modular homomorphism.