Mower will run fine until i engage the mower blades

You mau need to change your belt

jack up the mower and see if the blades will turn by hand freely. If hot you could have something stuck in there or a seized bearing.

sounds like wrong belt you raise it belts tighter lower belts looser or the tensioner has come loose

Hi, first charge or replace the battery if necessary then check that the mower is charging the battery. The pto is probably fine but it needs battery voltage to engage. If the mower has stopped charging the battery may have had enough voltage to hold the pto in until you disengaged it but not enough to re engage it. Regards Phil.

clean the jets inside the carb

The engine in this machine seems to be powerful. If you can remove the belt from the pulley that drives the blade. Try turning the pulley buy hand. If it turns tight or not at all this is your problem. The unit that is the pulley to blade assembly is seized or ready to seize. Sometimes they are designed to be taken apart and repaired, and sometimes they are not. Did you check for a grease fitting within the pulley to blade unit?
I would like to hear how this works out for you. Best of luck.

A common problem on the 21 & 23HP models is bent valve push rods on one cylinder. With this the engine runs and seems fine, albeit a bit quieter, until you put a load on it like engaging the blades which causes it to nearly die. So you may want to eliminate this as a possibility by running the engine with one plug wire removed and then the other. If it runs with one wire on alone, but not with the other wire on alone, then you have a nonworking cylinder even though it is receiving spark. Just a thought.

Check throttle cable at carb and make sure you have full travel. Sounds like rpm is little to low. Throttle cables are usually held in place with 1 screw and a washer. Put cable at full throttle loosen screwand pull back until throttle at carb in at full then tighten screw.

Backfiring is a sign of too rich on gas and not enough air. Not enough gas will cause the mower to bog down when blades are engaged as it isn't powerful enough to run the motor AND turn the blades. Check your carb and see if you need to clean it and replace diaphgram. Sounds a lot like a carb problem.

check your fuel pump. Fuel pump problem.