OK Muhydeen, let's correct you.
The two example sentences where you wrote "can" and "not" side by side have nothing in common with "cannot":
I can not only sing but also dance. I can not only teach mathematics but also economics and English.
In these sentences the "not" negated the exclusion expressed by the word "only" rather than the possibility expressed by “can".

Dear dbfreak,
"Can you not shake your leg when I’m in the room? You can just not shake, ok? -> You can not shake it."
First, you probably debate a permission to shake my leg when you are in the room, which should be expressed with the word "may" rather than the physical possibility expressed by "can". However you are correct to state that your example is also correct grammatically. Only then it means that you are possibly so huge as to make shaking a leg physically impossible by your presence in the tiny room. Alternatively, you could be one scary ... whose presence paralyses the poor victim and he or she can not move the darn leg.