He made a personal visit to Cuban leader Fidel Castro whom he greatly admired, when the Argentina authorities would rather that he did not.(original)
Question:
Is the original grammatically correct?
I greatly grammatically doubt such as
"the Argentina authorities would rather that he did not", which I don't know how to understand it by grammar. Could you tell me the meaning of it, and the reason why it is written that way just by grammar? and better to give other examples in the similar grammar or rule as above

Question

He made a personal visit to Cuban leader Fidel Castro whom he greatly admired, when the Argentina authorities would rather that he did not.(original)
Question: 
Is the original grammatically correct?
I greatly grammatically doubt such as  
"the Argentina authorities would rather that he did not", which  I don't  know how to understand it by grammar. Could you tell me the meaning of it, and the reason why it is written that way just by grammar? and better to give other examples in the similar grammar or rule as above

Answer

Do you want to say that the authorities right now don't like that he visited, or are you saying that the authorities didn't like it at the time of the visit?

Answer

I would use “had not” instead of “did not”, that should fix the issue.

Answer

I agree “had not” is better.

Another example:

I would rather he stopped smoking. (Present) I want him to stop smoking.

I would rather he had stopped smoking (Past) I wanted him to stop smoking. 
This could also be:
I would have preferred it if he had stopped smoking 
I would have preferred him to stop smoking.