This is probably a very naive question from a field that I don't have much background from, but a combination of curiosity and the fact that conceptual questions get very good answers here on MO seemed enough motivation to ask it.
Solitons are very interesting objects for a number of special properties they have, such as being "particle like", being stable and conserving their geometric features. Some of these properties are related to symmetries related to the notion of solitons, symmetries whose presence is mysterious to me.
My question is, therefore, what is a conceptually satisfying explanation of the non-obvious symmetries of solitons? Where do they come from and/or what causes them? I could not even perform a (fruitful) basic search since I don't know even the key-words necessary for it.