Hi Ralf,
JAP made a lot of engines, and as Rex points out not all of them went into motorcycles. By 1927, when your engine was built, there weren't many makers using the 500/550/600cc side valve singles. Possibly your engine came from a continental maker, rather than the UK.
Anyway, I don't really understand LY/I as a JAP prefix. The L is OK (85.7 bore) but Clew says that Y means twin port head, in which case I'd expect Y to be always accompanied by O (eg LOY) meaning ohv. Not to worry. LY probably means 550 sv. I after the / is unambiguously 1927.
Here are some sv engines form the 1927 JAP catalogue. Do either of them look like yours? Your photos don't show this side...
Cheers
Leon