As a retired Police rider for some 14 years I rode Nortons, BMW 750, 800, and K100's and all of them weaved to some extent.
The Nortons had isolastic mountings that allowed the rear end to feel like it was doing cartwheels
The BMW twins suffered from the the effect of the engine being horizontally opposed and we were warned not to change gear going round corners (who would want to?) and the effect of the 'barn door' fairing in turbulence.
The K100 was an absolute pig at 120+ and I've used both lanes of a dual carriageway to sort it out in the past.
All have had full police equipment of radios panniers and lights which I'm sure played their part..
Police workshops, in my time, were pretty good and no expense was spared to fix any problem put into them.
Since leaving I worked for Honda as a MAC Instructor for 5 years and went through 5 Pan 1100's and found them to be superb. I also worked with several Honda Garages taking out riders on test days and rode the 1300 many times at (ahem) a 'brisk rate', and found only 1 that handled badly with the front washing out on bends at various speeds. I immediately refused to ride that particular machine again, and put it into workshops for investigation.
I have now bought a brand new 1300 and within 3000 miles replaced the front and rear suspension with Maxton units, this has a stronger spring on the rear and uprated internals, the front is also more controlled and doesn't dive on the brakes as per the standard. I have absolutely no weave nor problem with the bike. I have already posted Maxtons report on the 1300 but I have pasted it below for info:
HONDA PAN EUROPEAN STX1300 2003 to 2007
FRONT
The springs are on the soft side which can cause the forks to bottom out under heavy breaking, this depends on your rider weight and again how hard you ride the bike, the big problem is they do not have enough rebound damping, which makes the forks spring back very quickly when you let go of the brake or accelerate, this causes the bike to run wide out of corners and makes the bike difficult to turn in to a corner. We modify the forks fitting harder springs to suit the type of riding you do and also your rider weight, we also revalve the forks increasing the rebound damping to control the forks.
REAR
The standard rear units on the new Pan European are too soft. They do not have enough damping to support the bike and the spring is not strong enough either. A lot of people complain about the new Pan European having a "high speed weave" or "lack of stability", this is partly due to the shock absorber being too soft. Unfortunately the standard unit can not be resprung and all we can do to help cure this problem is replace the unit. We supply a replacement unit for this bike which is an Alloy bodied high pressure gas unit and is fully adjustable via a shim stack system, the unit is built to order so is sprung and valved to suit your rider weight and what you use the bike for.
I have lost friends within the job to accidents, both cars and bikes.. I offer my sincere sympathy to the family of any those concerned.
Steve