I'm having the same problem with my BX2200.
I replaced the stop solenoid - no go. Took the ignition switch apart and cleaned the contacts - still nothing. Same with checking the fuses. The other symptom I'm having is I don't hear the fuel pump or have the battery and coil lights lit when the key is in the on position.
On the upside, it runs fine with the stop solenoid removed
Any ideas out there?
Thanks.