OK MIlt, I think we are finally getting somewhere.
To quote from your most recent post.
"Your situation 1 is now real close to how it happens except for that increased use of rudder. Before you put on that extra rudder we were balanced having arrested the yaw.
So now you go and spoil it all by changing the rudder. The tiniest bit either way will unbalance us again. But if all you want to do is go along your original track/course then by all means use a little rudder to do a slow flat turn to do just that. It is only 2 or 3 degrees anyway. Having turned to the new heading you must return to balance again by returning the rudder to its precise original anti yaw position. Remember if you change CAS or thrust or drag you unbalance and there will be a changed rudder deflection to regain balance."
I did not say that this manoeuvre was a good idea nor that it is easy to achieve, but simply that it is possible. Now let's be clear about it, we are talking about a situation in which the aircraft is sideslipping to the left. YES or NO?
And:
"Situation 2. Yes that is what we do to get the rudder back close to centre. Usually about 5 degrees thank you which gives us enough horizontal component of lift to cause us to sideslip to the desired extent.."
And isn't this sideslip to the right? YES or NO?
Unless I am mistaken we now have agreement that it is possible to sideslip either to the left or to the right with a left engine out, depending on the combination of bank angle and rudder.
Now we can get back to my original argument.
If we can sideslip left and we can sideslip right, then presumably we can gradually change from slipping left to slipping right. And at some point during the manoeuvre we will not be slipping at all.
If you look at the situation you will see that this will occur with a slight bank towards the live engine. This means that it is possible to have zero sideslip when banked towards the live engine in asymmetric flight.