Battery charge, get hold of a voltage meter and put it accross the terminals. Positive terminal might be red, but also identified by the way it is routed directly from battery to starter motor and also routed to the relay assy. Negative terminal will be routed direct from terminal to engine casing (grounded). Volt reading accross the circuit will optimally stay around 13Volts with the engine off. It should read a higher voltage, 13.8+ is common while the engine is running, with lights on (recommend you leave the lights on always as it take some voltage away from the regulator). If the bike's been sitting unused or has a poor charge from the magneto when running, the power will drop, it doesn't take much either, even a reading of around 12V is low enough that the starter will keep turning over but won't have the punch to start her up.
A common recommendation/solution is to kick start from cold and use the Lecky as a re-start. (I believe this is the design specs of the Elec re-start anyway) The initial kick starting will save battery power, and the riding done thereafter will charge the battery more for when you Elec re-start- providing the magneto is upping the voltage in the system once your engine's running.
Next - use a reasionably cheep 12V car or bike battery charger, nothing heavy duty, no truck/forklift battery chargers! Charge for about half an hour at a time.
If you've just bought the bike and it's been in a shop for weeks, or the previous owner hasn't been regularly using it, the batter may well read 12V+ but it's going to take a decent ride/charge to bring the battery back to life. It's not common either to charge your stale Berg and half an hour later the battery reads 13.5V but then it will only withstand 2 Elec starts before dropping charge.
Good luck I hope it's a simple fix like this.
Can you tell me about the 501 on the street. I'm giving thought to getting motard wheels for my Berg for the street but what's it like to ride for a while. I presume the single cylinders give alot more vibration than your 4cyl ZX6. And obviously the top end speed of both bikes is at different ends of the scale- does that make the berg seem like a close ratio box? When I'm riding off road the 6sp FE box is not close ratio but on open roads it's a different story. I think it looks like fun, and the riding position of a motard appeals more than a sport bike to me. Give me some feedback about having such long travel suspension on the road too.
Enjoy your riding