Starts up, then goes dark

my good old iMac G5 has developed a new problem. When turned on it starts up quite normally but after a short while, either towards the end of the boot process or even after login the screen suddenly goes dark. No way of reviving it except force shutdown and then boot new.

When used as external HD via Firewire target mode the disk is accessible without problems, even for extended periods of time.

My theory would be that either there's a problem with the GPU or with the PSU. Apple doesn't offer repair or spares for those legacy models anymore. So any ideas on how to find out what's causing the problem and repairing it would be great!