Ok,
I did some digging and the PA420A engine was fitted to the JD F510 mower
The PA420A engine is fitted with an external trigger module , called an Igniter for some stupid reason known only to Kawasaki
Now magneto ignitions theory 101
When a magnet approaches a coil it will generate a voltage in that coil.
then as the magnet departs in will generate another voltage with the opposite polarity
Depending upon which way the maget is fitted or the coil is would will determine if you get a posative pules first followed by a neagative one or visa versa
To make a spark you have to suddenly collapse a flux field within a coil by grounding it. at its highest voltage ( remember it is AC so it can be + or - volts ).
To prevent any spark being generated by the coil it gets grounded all the time.
Decades ago this was done with a set of points that grounded the primary winding plus a switch running in parallel with the points .
Points do not care weather they are working on the + pulse or the - pulse .
In 1966 a clever Hungarian immigrant living in OZ worked out he could replace the troublesome points with a solid state trigger using the rising voltage in the coil to cause the coil to ground at the right time.
This was called the Hall effect trigger and was marketed by them as the Atom ignition module. A stand alone solid state device that you simply replaced the points with.
However this device IS POLARITY SENSITIVE .
After patient had expired so they would not have to pay the 50¢ royalty every engine maker that had not already switched to solid state ignition did so by fitting their own module.
Then the engine makers realized that they could incorperate this trigger chip within the coil and charge a small fortune for them.
Now your engine originally had a stand alone trigger ( igniter ) module with a plain coil.
So to work it has to be used with a plain coil not a latter one with a module included .
I fear that your problem is you have non compatiable magneto parts or a dead igniter module that you are yet to locate
So follow the kill wire it should connect to the igniter module and it has to be there some where unless the coil has been upgraded to one with an inbuilt module.
You might find this Video form Taryl Fixes all helpful.
Note to work you have to install the coil upside down to get to polarity of the generated voltage correct for the module to work
Kawasaki igniter coil ignition modification.