Some tips that might be helpful here. First of all, beware of single cylinder riding mower engines, new and used. I'm not saying to disregard them, just be aware of some problems that are common to them. Single cylinder engines have a longer piston stroke that produces higher torque, which is good for more hp and torque with less cubic centimeters, but they also have a compression release on the camshaft assembly that is well known to "grenade' and if it does, you lose the necessary compression release feature. Sometimes it destroys the internals and sometimes the engine will be beyond repair. When that happens, the starter is unable to turn the engine over and it will not start. Sometimes loose valve adjustments cause the same symptom. The compression release at starting rpm (slow engine turning over) normally engages a valve lifter and causes it to "wink "a valve open, temporarily relieving the compression pressure on the long stroke higher compression engine. When started, and up to speed, the compression release moves away by centrifugal force so that the engine can take advantage of full compression when running. A valve lash that is too loose, when properly adjusted, (should be .04" on the intake valve lash setting and .06" exhaust valve lash setting) will fix the problem if the compression release is operating properly. Sometimes, if you have a strong high cca battery and solenoid and good electrical connections you can back the engine up by hand to let it get a "running start "but more often than not it still won't be able to overcome the higher compression problem. The only solution is cam replacement or engine replacement which can get expensive. Also, these engines require a high-capacity heavy duty starter, so beware of replacing parts until you "stumble onto the perceived problem". If the starter is replaced with a standard duty starter and/or a lower cca battery you can have the same symptom. First, remove the valve cover. Then while slowly turning the engine over by hand look for a "brief wink" at either valve. Second, check the valve to rocker arm clearance with the engine cold. If the gaps are wide, adjust them and turn the engine over slowly by hand again. If there is no wink there is no compression release. Also check for a mouse nest under the engine cover on the topside of the cylinder fins. That will cause overheating and movement of the valve guide (usually the exhaust valve) which in turn will bend a push rod also causing loss of compression release, or worse. Most two cylinders, to my knowledge, do not have a compression release and if not, there is no chance of losing it. My experience is that when 2 cylinders act like they have no compression release the starter is weak, the battery has insufficient cold cranking amps or has discharged, the solenoid has carbon buildup, the battery charging system is faulty, and/or cables and connections Positive or Ground are corroded. Check all of these before replacing parts. Poor electrical connections will heat up when engaging the starter. Loose engine mounting bolts on oil covered engine mounts can cause loss of a good ground. If the starter will not turn over (drags) attach a jumper cable to the negative battery terminal on one end and the other end somewhere on the engine casing. use only one cable lead, negative preferred. You are simply grounding the engine and the negative terminal of the battery to prove that there is a good ground. If the engine then turns over properly you have proven a loose ground connection. Check the engine mounting bolts first and all other negative large wire ground connections.
Mowers with electric clutches have a higher alternator charging capacity to recharge the battery after starting and also provide sufficient current to keep the electric clutch fully engaged. NEVER JUMP START YOUR ELECTRIC CLUTCH EQUIPPED MOWER THAT HAS A DISCHARGED BATTERY!! It can potentially not only burn up the voltage regulator from overloading it but also cause the clutch to overheat and burn up. That is around a $250.00 to $1000.00 repair depending on the shop charges and parts. Replace the battery if it is discharged, it is a very inexpensive repair in comparison. Be sure to select the proper CCA battery for your mower (The higher the better that fits the mounting dimensions of the old battery). Mowers with lever engaged blade mechanisms have a lower capacity charging system that uses a rectifying diode that acts as a voltage regulator by "clipping" the AC current enough for the battery to receive it for recharging. If you accidently hook up your battery backwards Plus to minus it may burn up the diode and your mower will still start and run but the battery cannot recharge so after a few mowings your battery is dead again. You then return the battery or buy a new on and the cycle repeats itself. A simple check is to test the battery voltage when replaced, record or remember that voltage, start and accelerate the engine while still reading battery voltage and the voltage should inch up until around 13.8 volts. If no voltage rise the charging system is faulty. A brand new fully charged battery may not immediately see a voltage rise so be patient. The diode is normally located in the wiring harness coming from under the engine cover on the right side of the engine. You should see a cylindrical hump about 3/8 inch long under the wire insulation. That is the diode. I use a 6-amp rectifying diode to repair it. A good diode will pass current in one direction but not in the other. There should be a silver band on one end of the bad and good diodes. The bad diode will not pass current in either direction. Do an ohms measurement to determine whether it allows a reading in one direction but not the other direction. Clip out the old diode and solder in the new. It is much easier than replacing the entire harness.
Gotta Go! I hope this helps.
By the way be safe always. Think it through before attempting any machinery use or repair, know your limitations, and get understanding first.