the waulking song lyrics don't appear to be about a swan. The Skilda recording is on youtube.

Here is a slow air called Not a Swan on the Lake
The first tunes in these medleys.
https://www.youtube.com/watch?v=ZjPI17XYtUg

https://www.youtube.com/watch?v=rV6RR_ct9YMhttps://www.youtube.com/watch?v=rV6RR_ct9YM

sheet music: http://www.therealviperpiper.com/viewtopic.php?t=1503

I'm trying to think of what song/s I've heard to a similar air, but I don't think I association that slow tune with the chorus "air fàillirinn".