Took Harv's suggestion to heart and did a Safe Mode restart followed by another restart and the Repair Permissions thing. This morning, tapped the space bar to wake up the box, and lo!, and even behold!, the screen lights up, with NO ALERT. But, wait for it. I click on the phone icon to get to "Connect", 'cos I want to talk to you guys, and what happens?
That darned "Restart your computer" alert comes up.
Now,
In answer to
Gary S, no, nothing new other than loading the driver for the ADSL modem (Speedtouch) and loading Virtual PC for Mac.
airbusdriver I've now downloaded MacJanitor and run it for "All tasks", so we'll see if that does the trick. And thanks for the suggestion. Will let you know in a day or so.
Meantime, a big than you to all for trying to help