30-08-15, 12:58 AM
The gear selector on the FZS600 has a design flaw. Once the claw on the selector arm is a bit worn it struggles to pull the drum round far enough for the gears to engage cleanly. This particularly affects 2nd gear, but can also prevent 1st from selecting (the drum needs to be pulled round a little further for 1st and 2nd because of the neutral detent between them).
The solution (for me, anyway) was to grind a small amount of metal off part of the selector arm to allow it to rotate further.
It sounds to me as if your bike's in the early stages of the problem and so, with luck, you won't need to replace the selector forks and drum. Details are here (link takes you to the middle of the thread since the first bit is long enough that you'll have dozed off by the time you reach the useful section).