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Are you getting any malfunction codes set in the engine control computer or have you checked for codes? When you say that it will not stay running is the idle just extremely low or when you started it it immediately stalls? If you push the accelerator pedal does the engine rev up and stay running?
Based on what you're telling me the alternator is not charging. Did you have the alternator tested? Even this new one that you installed? When the engine is running are you getting any warning lights on the display?