Well, apparently Freddie made a cover of Frank Sinatra's "New York, New York" for use in the movie. There are rumors of a demo recording or full-on track of Freddie singing the whole song, but to my knowledge, no such thing exists (except in fake demos and stuff like that). All that officialy exists is that one-minute snippet that you hear during the freaky driving scene with Kurrgan and Brenda.
I like how, in the movie, the song goes from Don't Lose Your Head to New York, New York. And if you listen to the song closely, you can hear that it keeps the same beat and tempo, which is nice. Almost like a medley :D
We can only grow the way the wind blows
On a bare and weathered shore
We can only bow to the here and now
In our elemental war
- Rush, "The Way The Wind Blows"