  $J$-holomorphic curves in nearly K\"ahler $\mathbb{CP}^3$ are related to
minimal surfaces in $S^4$ as well as associative submanifolds in
$\Lambda^2_-(S^4)$. We introduce the class of transverse $J$-holomorphic curves
and establish a Bonnet-type theorem for them. We classify flat tori in $S^4$
and construct moment-type maps from $\mathbb{CP}^3$ to relate them to the
theory of $\mathrm{U}(1)$-invariant minimal surfaces on $S^4$.
