Keil & Delitzsch, on Judg 5:1

“In that day Debbora and Barac son of Abinoem sung, and said:”

“The historical introduction ("Then sang Deborah and Barak the son of Abinoam on that day, saying") takes the place of a heading, and does not mean that the song of Deborah and Barak which follows was composed by them jointly, but simply that it was sung by them together, in commemoration of the victory. The poetess or writer of the song, according to Jdg 5:3, Jdg 5:7, and Jdg 5:12, was Deborah. The song itself opens with a summons to praise the Lord for the willing and joyful rising up of His people.”

Imported from an open dataset — not yet checked against the printed edition.