Pre-existing betatron motion and spin flipping with RF fields in storage rings

It seems to be a common perception that pre-existing, non-spin-resonant, vertical, betatron motion in storage rings has little influence on spin-flipping with external, resonant, radio-frequency, magnetic fields. While this is often the case, care is needed, and useful insights are obtained by studying the geometry of the invariant spin field (ISF). I illustrate this by numerical means for protons in the COSY ring and I suggest a test that could be carried out at COSY.