Swan river (4)

I believe the answer is:

avon

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This clue might be a double definition.

'swan' is the first definition.
Both the definition and answer are singular nouns.
Maybe there's an association between them I don't understand?

'river?' is the second definition.
(I've seen this before)

Can you help me to learn more?

(Other definitions for avon that I've seen before include "Bath's river" , "River running through Bath" , "English river flowing through Stratford" , "River and former county" , "River at Shakespeare's Warwickshire birthplace" .)

I've seen this clue in the Evening Standard and The Irish Independent.

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