The Filipino combative arts integrate knives in general. The balisong is just another form of knife.
One important characteristic of FCA knife techniques is the ability to use the knife in any grip position and to change grips on the fly. For example, you can slash one way in a forward grip, reverse the grip and come back in a reverse grip all very smoothly. Some knife designs don't lend themselves well to this. So-called knuckle knives are an extreme case of a knife rather non-optimal for FCA techniques. A balisong, on the other hand, is very well suited for such techniques.
Another very important component of FCA is the use of striking weapons and of destructive striking techniques. A closed balisong is an excellent impact weapon.
A closed balisong also works well to reinforce joint locks which are another important component of FCA.
So, the balisong fits well into the Filipino combative arts. This does not mean that it won't fit into some other art, but it does mean that you're probably more likely to see it taught in an FCA class.
At API, where I study, basic combative manipulation comes up as an aside every now and then, but artistic manipulation is generally left the interested student as an independent exercise, probably with my help.
Perhaps Donna will chime in here and share her much more diverse experience with us.
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Chuck
Balisongs -- because it don't mean a thing if it ain't got that swing!
http://www.balisongcollector.com