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2nd Edit: So far, I see that a possible fix is to bake the motherboard or do a solder reflow of the graphics card chip on the motherboard, is that correct?
The short answer is no, Older versions of thinkpads with ati graphics suffered from detached chips due to flexing and the solution was to reflow the solder. When the T61 series displayed gpu issues shops tried the same fix and at first it seemed to work, but it didn't take long to discover there as a 100% failure rate on the repairs because the initial failure wasn't the solder, rather caused by microfractures within the chip due to stress from the bonded materials that expand at different rates and after many heat cycles the material would fracture along with the circuits and as soon as a vital circuit was damaged the failure occurs. When reflowing or baking the board the heat moves th circuits can can cause temp contact across the fractured circuits and the allusion of it working. This has lead to a nightmare for thinkpad fans worldwide was disrebutable people used this method to temp repair laptops and/or motherboards so they could be auctioned online. At one point I calculated 40% of the members I spoke to had a failure within a couple months of buying the laptop online and had no recourse. Honestly I wish people would stop publishing these so called "solutions" and it only results in a resurgence of these scams. The only way to fix this is a new motherboard. Finding one that's never failed is an option, or they did make an improved version of this chip that rarely fails but it was only used for about a month before the series was discontinued. They are extremely rare and I can sometimes get them which I've done for several members, or you can swap the nVidia board for one with Intel graphics. I'd be glad to help with this or you can source one yourself. There is also the option to replace the board with a T60 board on some models, they are pretty cheap, but other parts are required and some modifications depending on the model.
In theory it's possible to replace the chip, but nVidia hasn't sold them retail since 2008 and and up until about 2010 they only made them for OEM partners to fulfill contractional obligations since the cost of producing small runs of chips always loses money. The work involved in setting up a production line requires millions be made for it to be profitable. There are also fake chips coming out of china that are usually old chips with the numbers/dates polished off and new ones etched onto the die. Electronic chip salvagers have been doing this for decades in china and for the most part it wasn't a huge problem, but when defective chips are retagged as good ones, that is a huge problem just as when the USA dept of defense was sold thousands of military grade chips that were really just used post consumer chips that were re-badged and sold to greedy contractors at very attractive prices. There were also some makeshift factories refurbishing this boards with fake gpu chips in southeast asia, the most reasent were maylasia and somewhere in china, but I think they were also made in vietnam and/or thailand. In fact one UK based reballing company that was well known and respected in the thinkpad community was forced to shut down and liquidate his business after buying and selling some very official looking nVidia chip from reputable companies
If you're determined to attempt one of thise methods I'll warn you that if you bake a board hot enough to reflow the solder you will destory the board. Shops that do this type of reflowing/reballing have infrared reballing machines that cost as much as many cars and are run by computers that monitor the temps on the board to assure only the chip area gets hot enough to melt the solder and even if you managed to do it without making things worse and the computer appeared to improve, close examination usually shows the chip will have hot spots when stressed and I don't think there has been any reports of a reflowed chip that lasted more than a few weeks, maybe a few months in some rare cases but often if it works at all it won't even last a week.
An Intel board isprobably your best option, but I recommend you read the last 5-10 pages in this topic and the much larger topic regarding the T61p with one long, two short beeps and black screen. It's the same problem but the T61 fails less often and many times only partially fails as yours has where the t61p often fails completely, often with no warning, probably because the nvs140m graphics chips is actually a declocked Gforce chip evidenced by the fact that they can be overclocked upto 70% without showing any abnrmal stress, but that's not recommended if you have one of the defective era chip.
ThinkPad W-510 i7-820QM(1.73-3.06GHz) Quad Core... ThinkPad T500, T9900, 8gb SSD...FrankNpad T-60p/61p (X9000 2.8ghz) 8gb SSD ips FlexView...ThinkPad T-61p (T9300 2.5ghz) 8gb ram...Thinkpad X-61 Tablet 8gb ram.