\rangle$, where $r$ is cyclically reduced, be a one-relator group and let $M = F(S),N=F(T)$ be Magnus subgroups of $G$, allowing $M=N$. For any $g \in G$, either $gMg^{-1}\cap N$ is cyclic (possibly trivial) or $g\in NM$.

A simple argument also enables one to describe the form of an intersection $gM{g^{\prime}}^{-1} \cap N$, where $M,N$ are Magnus subgroups and $g, g^{\prime} \in G$, in terms of the intersections $gMg^{-1} \cap N$ and $g^{\prime}M{g^{\prime}}^{-1} \cap N$.

It is surprising that the questions addressed in and have not been examined more extensively, given that some of the difficulty in studying one-relator groups arises precisely from the situation where a pair of Magnus subgroups have *exceptional* intersection, that is $F(S)\cap F(T)\neq F(S\cap T)$. However there are some partial results that deal with special cases of and . In particular Bagherzadeh [@Ba] has shown that if $M=F(S)$ is a Magnus subgroup and $g \notin M$, then $gMg^{-1}\cap M$ is cyclic (possibly trivial) and in [@Br], Brodski[ĭ]{} actually considered a more general situation and showed that in a one-relator product $\langle A * B \ |\  r=1\rangle$ of locally indicable groups, the intersections $A \cap B$, $gAg^{-1} \cap A$ and $gAg^{-1} \cap B$ are all 