If not electrical issue what about a vapor lock created by carb problem? Swear it was cheaper to pitch. Since it had a non-adjustable carb at possibly $40.00 bucks to experiment. Theory applied though. When cooled down with water it would start but not run long. Even replaced coil and then stupidly never properly tested electric. Grab that coil wire and spin it. Can you dance without a partner? Got another engine from a similar model that the deck was messed up on. Swapped it out. Robbed old engine of new parts and other usable items and recycled the steel. Taa Dah?????? Maybe for you too??????
Clean out the carburetor with gas, the reason it wont keep running is its not getting enough gas through the carburetor.
Your flywheel key might be sheared
check your coil out
Clean the carburetor with special attention to the high speed jet and fuel passage.
If its less than 20 horse power, it probably wont have one. Mine is a briggs and stratton 14.5 hp and it does not have one.
The solenoid may be bad. If you need help testing the starting system contact me through my board and I will try to help.
carb. adjustment, fuel filter or the engine is shot.
two stroke will make it quik but its sore on petrol/oil and it wont last long........it also depends on wat size.....a 2 stroke shouldn't be more than 35 hp..........4 stroke is good mayb 1 cylinder No stick with a Briggs and stratton 8 hp or Higher, mine has a 12.5 and it awesome for off road
Without knowing what engine or what kind of mower it is this is just a guess. But, it is possible that your flywheel and/or coil have gotten rusty and are not getting the conact that they need to make the spark. Try removing the cover and sanding the flywheel and coil. If it is a riding mower with a key switch there are many possibilities including anything from a bad switch to one of the safety switches.
check to see if it is getting spark . you probably have the "kill mechanism " installed wrong . It's probably grounding out the ignition system. The "kill mechanism" is the cable operated mechanism that goes from the handle with a cable to the pull start housing . Its either improperly installed or out of adjustment.
Because the engine gets too hot. Let it cool down before you start it up again. Poor water on it.
valve adjustment too loose compression wont release,set,in .002 ex .oo4 at a hair past top dead center,plugs out