In the movie, 'Rio Bravo', with John Wayne, there is a song by Dean Martin & Ricky Nelson, what is it's name?

Question

I have never been able to ID the name of that song. One refrain goes through my mind is "With my 3, good companions, my rifle, my pony and me"

More words....

"Gonna hang,

my sombrero

on a limb, of a tree"

Mostly Dean Martin sang this song and Ricky Nelson did some back up, but for a little while, Ricky Nelson did sing part of it.

Answer 1

It's available on the Ricky Nelson 4-CD box set: "Legacy".

Answer 2

The name of the song is "My Rifle, My Pony and Me"