Recently I have been studying semi-toric systems and almost toric fibrations. For the purpose of semi-toric fibrations I have been reading these notes https://arxiv.org/pdf/math/0210033.pdf. Specifically in these notes, the author talks about an operation that modifies fibrations, which is called a nodal trade. This is the content of lemma $6.3$, which under certain conditions essentially says we can in a symplectic manifold switch a focus-focus fiber with a fixed point, while preserving the fibers on a dense open subset.
Now these conditions are rather technical and so far I still don't have much intuition behind them or neither would I know how to compute them if I had a specific example to work with. Therefore I was wondering if anyone knows of a reference where we take a semi-toric system on symplectic $4$-manifold, which will give us an almost toric fibration, and we are able to check that the conditions to apply the nodal trade are verified? How would I be able to figure what an eigenray, the vanishing or collapsing covector are ? Can I do this just by looking at the image of the moment map?
Any insight is appreciated. Thanks in advance.
