The phrase 'shall not exist in the US' is not a complete thought, so it is not a sentence.
You need a noun, such as:
Racism shall not exist in the US.
Communism shall not exist in the US.
we shall speak of this anon.
I suppose I shall have to answer this question, eventually.
The word sertainly does not exist.
The word knowned does not exist. The words known and reknown definitely do exist.
Example sentence: We play after school.
Some people believe that fairies exist.
The word "mesovamemea" doesn't actually exist.
I can reassure you that the word 'reasure' does not exist.
I shall carry on, despite your ill will.
"I shall not dignify this insensitive remark with an answer"
I shall deliver the box on Friday.
Depending on how it is used, exist can be a verb. If the sentence was, "The frog exists," it would be used as a verb. Let's say the sentence is "I do not like to exist." In this case, the word "exist" is used as the object of the preposition, and "like" is the verb.