Tried to paste an excel spreadsheet into the last post. Did not work. I'll try again to post.
 
Engine                    Function     Inlet Valve Position     Exhaust Valve Position     Cylinder Position
Kawasaki FS541     Compression      closed                           closed                          TDC
                            Power Stroke      closed                           closed                        Bottom
                            Exhaust Stroke   closed                           opening                    Moving up
                            Intake Stroke     opening                         closed                     Moving down
                            Intake Stroke     closed                            closed                        Bottom

Kawasaki FH580     Compression         closed                          closed                           TDC
                                     ?                 closed                          opening                 Moving down
                                     ?                 closed                          closing                        bottom
                                     ?                 opening                        closed                      moving up
                                     ?                  open                            closed                           TDC
                                     ?                 closing                           closed                   Moving down
                                     ?                  closed                          closed                         bottom
 

Did you find a solution for this problem? I’m having the same issue with a FX691.

No I have not.
I ordered a new camshaft. When I compared it to the existing camshaft it is exactly the same as a new one.
 
If I could send a pic I could show the alignment of the crank and cam shafts.
 
The only other part that could be misaligned is the mating gear to the camshaft located on the crankshaft. This gear looks like it is pressed on. So in my simple mind it is possible for this gear to shift.
 
I keep going back to when I first rebuilt the engine. It started and ran for about 45-60 seconds then it quit. That's when I noticed the gas coming out of the carb. That's when i realized the mechanical timing was not correct.
 
I am going to order a new crankshaft and compare to existing one.
 

PASTVN1 there is a video (https://www.youtube.com/watch?v=kYNsDVJbN-c) that shows that the timing mark on the crankshaft for a FR691V engine. If you look at 29:30 on the video the timing mark for the crankshaft is 180 degrees from the keyway. Is your engine the same?  I noticed on my engine that the timing mark is not 180 from the keyway. Just curious if this is the same on these engines.

That secondary gear on the crank should come off easily.  
 
I just replaced that gear on a Briggs PW in which the alignment pin sheared off.
The new gear was now steel, not plastic...
 
.

Finally found a solution to this issue. The crankshaft gear is not keyed or pinned. The gear rotated by 5 teeth in reference to where the timing mark should be. The timing mark on the gear is 180 degrees opposite of the key way on the shaft. The left crankshaft is the correct location of the timing mark.

After installing the new crankshaft, the engine started immediately and ran for about 30-45 seconds and then it quit. I checked compression and cyl #2 had only 30psi. I pulled the engine and again the crankshaft gear has rotated out of position from the keyway. Something is binding causing the gear to slip.
 
Any suggestions? piston/rings? Help would be appreciated.

Is that gear just pressed on W/O any keyway/pins?
 
 
.

Yes

So what can cause the gear to slip? The cam shaft or its components (piston rings tappets valves?) are binding. Not sure. Any words of wisdom?