Why does piano music sometimes have two right hand lines?

Sometimes there's a solo instrument (or voice) that should be performing the top one, typically they'll put that on its own staff.

Sometimes it's a duet for teacher and pupil - teacher gets the hard part and the junior student gets just the one hand to worry about.

Sometimes if they publisher thinks you can't figure out what part is the melody they will write it out separately so that you can put emphasis on those notes.

Sometimes it's just impossible to write out the rhythm properly without having to split it up. (written on the same staff though)

Sometimes it's notes belonging to the left hand that sneak too high and are written on the treble clef staff (again, written on the existing staff for the right hand though, not a separate one)

Can you find an example of this and post a link?

Martians ;) Sorry, don't know. For really good piano players, two players sitting on the bench?