How to find Independent Systems with either Federation or Empire minor faction inside

You can go to the galaxy map and filter for independent systems.

Once you filter them, you can select a system and go to system view. Inside you will see all the minor factions in a system.

Minor factions with allegiances will display the Superpower Logo. Otherwise it will be absent.

Generally the bubble is arranged in a specific pattern. Once you know the pattern it can be easier to find systems with different allegiances.
 
You might find the third party websites, like inara.cz or eddb.io, would be quicker and easier searching than trying to use the in-game functions to search for such things.

Suffice to say, there should be plenty of them about and not too hard to find. If you look around the Fed/Imp border region, you will probably find several that have both a Fed and Imp non-ruling minor faction. Or you might try searching the Powerplay space of a power that doesn't favour the government types typically found in the Federation and Empire. Try Archon Delaine space, for example.

For the Empire in particular, check out pretty much any Indie system on the fringes of the Bubble in the Down region of space; you should find opposition Imp factions there. Most of those systems were once Imperial but have been stolen away by anti-Imperial player activity. Back when the game started, that entire region was a hemisphere of pure Imperial blue. Now there's yellow dots all over it.
 
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