Hello, Keberaxu

If you are looking for a tune, here's a good site to know about:

abc.notation.com

I went to it and searched for Eileen Aroon. The site brought up 17 known tunes, and most of them fell into two categories. One is a tune which starts with two eighth notes and hops up and down some. The other starts with quarter notes and climbs straight up. There are 17 versions of two tunes because they are in different keys or have slight variations in the notes.

On the first or second page, there is a little player which lets you listen to the piece.   

I think if you go to the site and do the search, you will find the tune you are looking for.