The string an other is vanishingly rare in English. In contrast another is positively pervasive. I think it would be fair to say that the second has eclipsed the first to the point of making the first unacceptable, even though it is a grammatical string. Both an and another are members of the category of determiners, while other, on the other hand, is an adjective. There's no grammatical ...
There's a formula: another = an + other. Think of it as of an article plus the word "other" that have historically merged into one word. Grammar requires some article before "other book"; either "the" or "a." Depending on the context, you get either "You need to buy the other book" (if, for instance, the guy bought only the first book out of the set of two) or "You need to buy an_other book ...
Other (other + plural or uncountable nonspecific noun) You can say: 'If I had another five dollars I could afford it.' - Five dollars is single unit of money. I have to admit that this answer lacked more information previously. With numbers it is acceptable to use another, but there are exceptions.
This is an interesting usage note extracted from the "each other" page of Yahoo! Education (emphasis added). It answers your question. It is often maintained that each other should be used to denote a reciprocal relation between two entities, with one another reserved for more than two: thus The twins dislike each other but The triplets dislike one another. Sixty-four percent of Usage ...
"No other," answers A, but my stationer." Here the expression, as Baker remarks, seems strictly proper, the words no other having a reference to A. But if the stationer had been the only visitor, he [the person speaking to B] should say, "none but," or "no other than the stationer called on me this morning." This is the opinion of Baker.
According to this, as "other" is an adjective you must write "two other places", "the two other places". In this ngram you can see that this order is prevalent. There is a very small minority (the other two sides, the other two groups, the other two men) but for these three words this is nonetheless by far the preferred order.
Perhaps, though, other equivalent phrases can highlight the difference in usage if you replace "except for" with "with the exception of" and "other than" with "apart from" Then we have "Are there any vegetables with the exception of asparagus?" which is clearly wrong. So the two phrases have overlapping uses, but are not equivalent.
Since “each other’s” basically means “ (mutually) the other person’s”, and we wouldn’t say “the other person’s faces” (unless the other person is two-faced), it makes more sense to say “each other’s face”. Nevertheless, the plural form seems to be more common in modern usage, so it can hardly be considered incorrect.
