I have a CD of mostly medieval and renaissance music, titled Music to make Merrie, (English Heritage EHCD29, 2004), that includes one track called The Bear Dance. This is the air in question. The CD styles this dance "traditional", but I think it dates from the 1970s.
On the other hand, I have The Most Dulcimer, (Greehays CR70714) a CD of Jean Ritchie's (a.k.a. kytrad) which has a song called Mourning Tears. The melody is William Billings's (1746-1800) When Jesus wept, first published in The New England Psalm Singer (Boston, 1770) page 56 (in the lower right hand corner--a convenient space filler!) Yet the name of Billings nowhere appears in the sleeve notes or the disk. Possibly the reason was space. The label wanted to get everything on a single sleeve sheet, folded once, so there may have been insufficient room for detail about the origins of all the songs. But it is easy to see how someone would be misled by the scanty sleeve-notes into thinking that the air was a southern folk-air (rather than of New English origin), or was original to Jean, rather than to Billings.