The melody appears in Playford's first edition of The English Dancing Master (1651). It is perhaps associated with the play by Shackerley Marmion (1603-1639), his second, a comedy that was also entitled "A Fine Companion", staged by Prince Charles's Men at Salisbury Court and published in 1633.
It was also printed in Sharp's Country Dance Tunes (1909), and Williamson's English, Welsh, Scottish and Irish Fiddle Tunes (1976).
It was recorded on The English Country Dancing Master, vol 2 by The Telemann Society and Country Capers by The New York Renaissance Band.