Since it's been shown that moons rise and set relative to the circles on the ruins site getting the relationship between the moons ground track and the ruins on the existing sites will help us identify the particular relationship.
We can then use this offset on other systems with moons to identify a search path.
Basically we can get the great circle path of the moon
https://en.m.wikipedia.org/wiki/Great_circle
Then offset it relative to the orbital direction and identify the small circle that runs parallel to it on the side that would result in it rising at the correct circle and setting at the other circle.
https://en.m.wikipedia.org/wiki/Circle_of_a_sphere#/search
To facilitate that I've taken the liberty to create the following google sheet to gather the necessary latitude and longitude points in a given time. Using these I can identify the great circle paths between them and their points. And then offset those to get the parallel track for the ruins.
Once we gather this data for the existing sites we can better correlate the angle of the ruins to the moons so we can derive the proper angle and track on the other landable bodies with moons for a search for new sites.
https://docs.google.com/spreadsheets/d/1-fqcT1t1QylyadOa23_fgFIlbRy7QBr2G2M4gjxOKsA
For anybody who wanted to contribute I've fixed the perms.. sorry about that