  We use the geometry of the Farey graph to give an alternative proof of the
fact that if $A \in GL_2\mathbb Z$ and $G_A=\mathbb Z^2 \rtimes_A \mathbb Z$ is
generated by two elements, there is a single Nielsen equivalence class of
$2$-element generating sets for $G_A$ unless $A$ is conjugate to $\pm
\left(\begin {smallmatrix} 2 & 1 \\ 1 & 1 \end {smallmatrix}\right )$, in which
case there are two.
