No, the pattern is this: each other is equivalent to: and The sentence "Each other wants a car" does not follow the pattern " each other", and so the "each other" is meaningless. Even if you turned it around, to get "A car is wanted by each other", you'd still only have ONE thing so it STILL doesn't follow the pattern. The essential point about this is that the "things" are simultaneously the subject and the object of the verb - if one is the subject, then the other is the object, and vice versa.
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