Synonyms for Bear On (same or very similar meaning)
WordNet sense 1 (keep or maintain in unaltered condition; cause to remain or last):
continue, preserve, uphold, carry on
WordNet sense 2 (have to do with or be relevant to):
concern, pertain, refer, relate, touch, come to, touch on
WordNet sense 3 (press, drive, or impel (someone) to action or completion of an action):
push
WordNet sense 4 (have an effect upon):
affect, impact, affect, bear upon
From the ODE community, based on WordNet
add/edit