Mac Pro mid-2010 shuts down

Apple Mac Pro, 2.66GHz 6-core Xeon X5650 "Mid 2010" with 16GB DDR3, 1.0TB 7,200RPM HDD, 18X "SuperDrive", HD 5770/1GB

Mac Pro mid 2010s running Snow Leopard (and Lion) worked fine for years but shut down suddenly about a week ago. Since then will not stay on for more than a few minutes and sometimes will not turn on at all.

Sometimes there is a single red flash from inside the machine when power button is pressed. At those times the machine doesn't turn on at all.

What I have tried:

I started in Recovery mode and did First Aid but it didn't report any problems.

I tried pressing D while starting up (after removing all peripherals except monitor keyboard and mouse) but the Apple Hardware Test didn't come on.

Our local repair shop has a policy of not working on such an old operating system.

Any suggestions would be appreciated.

Thanks,

Michele