  We prove that if $G_\phi=\langle F, t| t x t^{-1} =\phi(x), x\in F\rangle$ is
the mapping torus group of an injective endomorphism $\phi: F\to F$ of a free
group $F$ (of possibly infinite rank), then every two-generator subgroup $H$ of
$G_\phi$ is either free or a sub-mapping torus. As an application we show that
if $\phi\in \mathrm{Out}(F_r)$ (where $r\ge 2$) is a fully irreducible
atoroidal automorphism then every two-generator subgroup of $G_\phi$ is either
free or has finite index in $G_\phi$.
