What is the difference between Part 135 and Part 129?
I'm not at all sure what you're asking, but maybe I can at least point you towards more information.

First, the parts you mentioned define operating rules for different flight types (I'm simplifying a bit here):


Part 91: rules for all flights in the US
Part 121: rules for flights by a US airline
Part 129: rules for flights by a foreign airline in the US
Part 135: rules for 'on demand' (usually charter) flights by a US operator


Any given flight will operate under one or more of those sets or rules, e.g. a US airline flight from New York to Los Angeles would follow the rules in parts 91 and 121; a foreign flight from London to New York would follow parts 91 and 129 (while in US airspace). Individual operators and airlines also have OpSpecs that are approved by the FAA and that can add more rules for specific situations.

And I'm not entirely sure, but it looks like 14 CFR 129.119(c) effectively applies part 121 to foreign airlines too, by requiring them to comply with the same rules as US airlines:


  (c) Each foreign air carrier shall conform to the practices,
  procedures, and other requirements prescribed by the Administrator for
  U.S. air carriers for the areas to be operated in.


Without knowing what your mysterious problem was (and I understand you can't share it), I don't think it's possible to say much more than that.

(By the way, "filing" is something different. Filing a flight plan is required for various reasons but they apply to all aircraft, even private flights. If I want to fly a C172 under IFR then I have to file essentially the same flight plan information that an A380 pilot will file.)