YES you have a fan!
The fan is controlled by the water temperature sensor which feeds the temperature switching unit in the electrical tray under the tank.
I can bet you that your problem is a seized fan.
The fan is on the back of the radiator. If you remove the fairing side panel, you should be able to reach it with your fingers.
Try to turn it. It should be very easy unless the brushes holders had melted.
The motor is not waterproof, and when you wash your bike and hose the fan the water will sit in the motor.
When you start the bike and warm it up, if the engine becomes too hot the fan will be activated. The water sitting in the motor will result into sparks in the brushes holders and the plastic will melt. There is no way to fix it and you should replace the fan unit (not cheap at BMW, but some for sale on Ebay).
THIS IS A GOOD ADVISE FOR EVERYBODY: DO NOT WASH YOUR RADIATOR WITH A PRESSURE WASHER. TRY TO AVOID DIRECTING THE HOSE TO THE FAN. I saw this problem on numerous K bikes.
Yesterday, I posted the starter electrical schematic, and it's interesting to see that when you push on the starter, the load shed relay will cut the power to the temperature switching unit.
I had read somewhere that the starter will not turn if the engine temperature is too high. I guess this is a wrong statement!
Here is the schematic of the temperature switching unit (
I am showing only the connections to this unit as some of the connectors are feeding other devices ie: The kill switch also feeds the starter relay):
Bert