I have a doubt concerning the following structure:
The Asiatic lions are in danger of extinction, as are the Asiatic cheetahs. (like the Asiatic cheetahs is also possible?)
Where is this rule explained:
as + verb + word
like + word
http://grammar.ccc.commnet.edu/grammar/conjunctions.htm#subordinating_conjunctions
Read the explanation of The Case of Like and As in the box below the lists of conjunctions.
So ... have you decided how you'll rephrase your sentence?
The structure you are referring to is called a comparative construction, specifically the comparative construction with "as" and "like." In this case, it is used to draw a comparison between two things, highlighting their similarity.
The rule you are looking for is explained in grammar under comparative structures. In English, we use "as + adjective/adverb + as" to compare two things that have equal qualities. For example, "He is as tall as his brother."
However, when comparing actions or behaviors, we use "as + verb + as." In your example, "The Asiatic lions are in danger of extinction, as are the Asiatic cheetahs." Here, we are comparing the danger of extinction for both Asiatic lions and Asiatic cheetahs. Replacing "are" with "is" in "like the Asiatic cheetahs" would not be grammatically correct in this context.
On the other hand, "like" is used to compare nouns, not actions or behaviors. For example, "He sings like a professional." In your sentence, using "like" instead of "as" would not accurately convey the intended meaning of comparing the dangers faced by Asiatic lions and Asiatic cheetahs.
To summarize, the rule is that "as + verb + as" is used to compare actions or behaviors, while "like" is used to compare nouns. The specific rule you mentioned is explained under comparative structures in grammar resources.