Howie in [@H] where he provides a detailed description of how the *exceptional* case can arise and generalises to the case of a one-relator product of locally indicable groups. In addition his methods provide an algorithm to determine for a given one-relator group and two Magnus subgroups $M$ and $N$, whether or not $M\cap N$ is exceptional and to determine a generator for the additional infinite cycle in the exceptional case.

In contrast to the situation for the intersection of two Magnus subgroups, the algorithmic problems arising from remain open. The difficulty appears to be caused by the case of two-generator one-relator groups. For both and , there is nothing to prove in this case, for if $G = \langle a,b \ | \ r=1 \rangle$, then the Magnus subgroups $M=F(a)$ and $N=F(b)$ are both cyclic. In the case of , the algorithmic determination of $F(a) \cap F(b)$ is provided by a procedure based on the Baumslag–Taylor algorithm for determining the centre. The methods of [@H] then yield a procedure for the general case. For the case of when $G = \langle a,b \ | \ r=1 \rangle$ one has to be able to determine, for a given $g \in 