No, everything until なく is part of the same adverbial phrase. “That (person/thing)'s thoughts never wavering, […]”. As far as what the indicator is…I’m afraid I’ve got nothing for you, at least in this case. It’s very unhelpful to say “you just identify the part that looks like an adverbial phrase, and anything that doesn’t fit in there isn’t part of it”, but that’s often how parsing sentences works. You get used to it.
Side note (going back to the first bubble), even though the “rule” is that broadly that いる is for people and ある is for things, it’s pretty common to use ありました for people when you’re setting up a story, especially in the case of legends or fairy tales. It gives it kind of a “once upon a time, there was a…” type feel. So the initial expository sentence isn’t “There are things that are unchanging”, but rather something like “Once upon a time, there was an unchanging entity”. It’s that entity who has the unwavering thoughts from this current bubble.
(Incidentally, いる is a somewhat recent addition to the language, which might explain this usage.)