you can clean it without removing the fixed cup, the main reason for removing fixed cups is to replace the entire bottom bracket if the bearing races are shot. If both cups and spindle are still good, just leave the cup in place, clean and grease the rest, and re-install w/ new ball bearings.
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1989 Schwinn Paramount OS
1980 Mclean/Silk Hope Sport Touring
1983 Bianchi pista
1976 Fuji Feather track
1979 raleigh track
"I've consulted my sources and I'm pretty sure your derailleur does not exist"