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The underlying concept is: The waterway was somber. "To flow somberly" would describe the manner of flowing, but there is no somber way of flowing or a happy way of flowing. Water can flow slowly, sluggishly, fast, but not somberly.

The problem of adjectives after verbs in adverb position is very tricky and it takes some time to get a feeling for this problem. The school rule "after a verb follows an adverb" is really only a school rule. There should be added in brackets: (But there are a lot of cases where an adjective follows, because the intent is not to describe the manner.)