Hi Gilles.
You may want to start browsing this very site pages:
> museum page about 31E:
http://www.hpmuseum.org/hp31.htm
> how to disassemble a Spice (= the family the 31E belongs to):
http://www.hpmuseum.org/disasm.htm
> how to self test the 31E (from http://www.hpmuseum.org/features/31ef.htm):
"Self test by pressing STO and then ENTER. At the end of the test, the entire
display lights to indicate success"
> service tips:
http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/articles.cgi?read=361
> *multimedia* (great pictures indeed :) spice repair:
http://www.voidware.com/calcs/spicerepair.htm
> looking for professional service:
http://www.fixthatcalc.com/
Hope this helps (to start...).
Best regards.
Giancarlo