The motion of the sun across the sky is a result of the earth rotating so a sun that is motionless must be a result of the halting of this rotation.
I'm not suggesting for even a femto-second that I think this could ever happen, but couldn't you also make the Sun appear to stand still by increasing the orbital velocity of the Earth?
Specifically if you make the Earth orbit the Sun in the same time it takes the Earth to rotate on its axis the Sun will appear to be stationary in the sky (I'm sure I read this in a sci-fi novel years ago - talking about an alien planet and star obviously).
Thinking about it for a second - which is probably more than it deserves
- this would probably only work exactly if the orbit was truly circular.
I wish I didn't know now what I didn't know then