What does the line look like? Is it a hard line of on/off pixels or kind of faint and an inch or two wide?
This sounds like the clamp circuit on the neck boards that comes on during horizontal retrace. When I was playing around with ways to get rid of the "right side of the screen is brighter than the rest when image is black" problem, one solution left me with a faint vertical line off-center. This is the point where the clamp turns off and it makes the video signal "bump" a bit resulting in the vertical white line. It happens during retrace, which isn't blanked, and showed up only when the image was dark or black.
Cary