To exist = to be: therefore, there are/exist two mechanisms....
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Both are wrong. "exist" is used on its own or "there is" or "there are" can be used. "there + exist" does not exist as a phrase.
The verb "exists" corresponds to third person singular: I exist, You exist, He exists. This added "s" at the end of the verb does not indicate plural.
Adding a final "s" to the verb "exist" ("There exists two mechanisms") seems to me a mistranslation of the Spanish "Existen dos mecanismos", which is plural.
So, "two mechanisms exist", or "one mechanism exists".
The be