"She does not have any brother."
Translation:Sie hat keinen Bruder.
You can say "Sie hat keine Brüder" for that. I think this isn't about the question of whether she generally has siblings; it sounds more like an answer to "How's her brother?".
But it does sound more natural to me (not being a native speaker) to say "She does not have a brother", instead of "any". I think.
I have no idea what a Pearson sentence is, but either "she has no brother" or "she has no brothers" would make more sense in English. For me, I could see "she has no brother" in the less literal sense of a reference to having disowned her brother. "She does not have any brother" is likely to result in a quizzical response.
This sentence is broken English which would immediately give one away as a non-native speaker, since "brother" must take the plural form when paired with "any" in English. We simply don't say "she does not have any brother." Instead, the grammatically correct English phrase is "She does not have any brothers" (or "She does not have a brother" or "She has no brothers.") "Any" always goes with a plural noun in English, unless the noun is mass/uncountable noun, i.e "She does not have any bread" is fine because we don't say "breads." Since we do say "brothers" as brothers are countable, this sentence is a grammatical failure in English.