expression, godan verb (-gu)
Literary or poetic expression for crying or shedding tears, often with a sense of pouring out one's tears.
彼女は悲しみのあまり涙を注いだ。
She shed tears in her deep sorrow.
涙を流す is the common, neutral expression for shedding tears, while 涙を注ぐ is more literary and emphasizes pouring out tears.
The expression combines 涙 (tears) with 注ぐ (to pour), literally 'to pour tears'. The exact origin is uncertain, but it is a set phrase used in literary contexts.