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I made a video of what is going on
hope it's clear and sound is good
Thanks

vimeo.com/137685181

At first blush it seems like you have a poor pilot system and you're either flooding or starving the engine during initial throttle.

Fuel injection isn't it ??, sounds a tad rough at idle as if not all 6 are firing, or could be the video sound, she's starving, are you getting full fuel flow to injectors ?, but what do I know, I only have carbies, sorry just throwing ideas against the wall to see what sticks

First I have to say I have no first hand experience with the fuel injection on these bikes, but a fair bit of experience with fuel injection on cars, so here it goes.

Glad to see that you posted a video of your issues. a pic is worth a thousand words, a video - priceless, NICE!!!

First off, I hear a chain noise that I don't like. Sounds like a rattle in the background. I don't know what cam chain tensioner you have on your bike, but before I did anything I would be checking the cam chain tension first.

That "squeak" sounds like an off idle vacuum leak. When you stopped the bike after the last ride, did it pop back through the throttle bodies? If so, it may have blown off a vacuum line or a cap off of one of the balancing tubes.

You told us you found rust particles in the fuel filter. There should also be pintle filters in the top of the fuel injectors. There about 1/8" in diameter and maybe 3/16" long. It's possible that they are partially blocked with micro particles that passed through the main filter. Most cars will start to run bad after 100,000 kilometers and then need injector service. The primary problem with low flow injectors is blockage of the pintle filters.

I'm pretty sure the ZN fuel injection system derives its sense of required fuel by measuring the manifold absolute pressure which in layman's terms is manifold vacuum, so if you've developed a vacuum leak, then the feedback signal to the computer would be wrong and the computer would be adding the incorrect fuel amount.

It definately sounds lean to me and that ties in with either air leaks or fuel blockage/starvation or both.
It sure is noisy so as Kawboy has said make sure the cam chain tensioner is working because if it goes it is disasterous. That cam chain issue has killed more 13s than anything else.

Great video. Kawboy provided an excellent analysis. Please follow his advice. He noticed the chain noise, which was not apparent to me until I listened very carefully for it after he pointed it out. I do not know if the Voyager ZN1300's had the same cam chain tensioner problems as the KZ1300. Kawasaki addressed the problem on the ZN1300 by adding an external oil line directly to the cam chain tensioner. Whether this helped, is unknown to me.

Rust from the fuel tank has always been a problem with the 1300's. It tends to stop up the petcock, fuel solenoid, and carburetors. Some of the rust particles are so fine that they can pass through some fuel filters.

You might try to clean the fuel injectors by using BG44K. It is the only injector cleaner I've found to actually work. It is used in the transportation industry with great success. However, it won't remove rust particles that might be obstructing the fuel injector inlet filters.