Axial wobble or binary star system
There is absolutely no possibility that our Sun is part of a "binary star system". Even the least luminous star (with a mass of about 8% that of the Sun) would have an apparent brightness and proper motion that would easily have led to its discovery. See for example Is Sun a part of a binary system? and  Is it possible that the Sun has a binary partner (the Nemesis Theory) that has eluded detection?