what is our Solar System called rather than "the Solar System"?
I've found "Sol System" and "Monmatia", too, but is there more to it?
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It's just called "the Solar System". (Plenty of places and objects have names like that; it's no different from "the Arctic" or "the Moon" or "the Sun".)
("Sol system" is an invention of science fiction writers; it has no general use outside some science fiction contexts. Anything else is going to be something similar, or general crackpottery of one kind or another.)
To answer your question succinctly, the Solar System also goes by the names: The Copernican System, The Heliocentric System, and The Planetary System, in addition to the ones you have mentioned. There aren't too many other names, actually, so just stick to Solar System since it's the most widely accepted.