36in 2007 Vizeo model: VW37L HDTV10A screen/power off problem

Sounds like a motherboard problem.
The best way to proceed will be to call the customer care center and tell them the symptoms. They should be able to give you a figure on the amount required for repair. Otherwise, they will send a technician who will be able to tell you the amount required for repair. You can then decide whether to repair or buy a new one. The Technician would of course demand a nominal amount for his/her visit.

Whether you should go for repair or replacement would depend on how old is your television. If it is too (<5 years) old then you can consider it a signal to get an upgrade but otherwise, a repair can be kept on the cards.

This appears to be a technical issue on the television set. I would advise you to consult the authorized technician to get the issue resolved. Get an estimate of how much would the repair cost be - that is if the TV is likely to get repaired.
Depending upon the exact problem that your TV is facing and the cost of repair, you can decide on further course of action. If the TV has grown quite old, maybe it is time you should decide to replace it.

This definitely looks like some motherboard and control mechanism issue. And adding to that it has the display issue too. We faced similar issue with the Panasonic and it was related to the control mechanism that feeds the display. So this can be one thing that you may have to get corrected with the TVs service center. If the TV is in warranty then minor motherboard repairs are free and you just pay for the service charge. In case of display issue and motherboard issue combined that may end up costing you a lot. So if it's new TV then get it replaced under warranty. If the cost exceeds the repair cost, then it's worth buying a new TV. But for that I think it'd be wise to discuss the cost and repairs with the technician. This way you can decide whither to repair the unit or replace it with some other model.