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Shift from first to third - second gear is much overrated Seriously, as bulletproof as the flying bricks are, I'd swap out the gearbox
When you say "2nd gear went out", what exactly is happening? Is it hard to shift into 2nd? Does it pop out of gear? If so, it could be a bent shift fork. If you are dependant on dealers and professional mechanics to keep your K75 maintained, then you will spend a pretty penny for this type of work.Parts for the classic K's are interchangeable so you can source an 86 or later K100 or K75 transmission as a replacement (stay away from 85 or earlier parts as the splines are different). That gives you lots of options on Craiglist, Ebay or through Copart. You can also think about going for a K1100 transmission, drive shaft and rear drive from a wrecked bike. For example, I found a complete K1100LT rear end for $400 and used it on my bike. Direct bolt on replacement, no fiddling with parts and I now have a Paralever K75.Swapping the transmission on the K is a weekend job even for a shade tree mechanic. There is plenty of help and advice available on K100-forum or Motobrick from the members there.I'd contemplate fixing it because parts for these motorcycles are still inexpensive and they are real appliances when it comes to maintenance. You can sell the defective transmission to recoup some of your costs.Check the URL in my signature below to see my K75 build blog.
I've had two K75s (naked K75 and K75RT for winter) for about 14 years and do all my own work, so I know a bit about them. In my opinion, nothing ever built by BMW comes close to their stone-axe reliability. I'd have a really hard time parting with either, since they always get the job done, if the "job" is trouble-free transportation in any weather.Wish you were closer; I'd be glad to R&R the trans for you, as I have a lift and special sawhorse just for K bike transmission spline lubes.If you were planning to repair the transmission, I'd strongly recommend that Tom Cutter in eastern PA do the work; he's an expert:http://rubberchickenracinggarage.com/index.html