Idle speed problems

[Kthiry]

Hello there,

i restored my lucas mill model 825 saw to working order but i have still two problems to solve.

So first, when my gas is off, most of the time, the engine stop itself instead of just hold the idle speed. The engine is not that old, its a kohler ch740. 

I thought maybe it was because the engine was not idling fast enough, is it possible ?  
If its that, there is no carb on my engine, instead of that this is a direct injection engine, so is there a way to adjust the idle speed or it is just set at the factory and i can't do anything with that ?

other times when the engine is idling, which is very rare, the blade still turning a bit and if i remember she is supposed to be completely stopped, any suggestion for that ? I heard about the clutch of the engine, do i have to change it maybe ?

Thanks for your help.

p.s. thanks DonP for your help with the pulley, its was all what i needed !