OK Here's my idea of how the A part might be completed plus an invented B part wich uses the same resolutions as the A part and then a C part which uses the idea that leeneia suggested, starting in the relative major key.

X:1
T:unknown
L:1/8
M:3/4
K:Em
E3F GA | Be e4 | e3 f ed | B4 AB |
cB AB cA | BA G2 B,2 | CE A2 G2 | F6 |
E3 F GA | Be e4 | e3 f ed | B4 AB |
cB AB cA | BA GA BG | B,G F3 E | E6 |]
e2 B3 G | c2 F4 | B3 A GF | GF GA B2 |
cB cB Ac | BA G2 B,2 | CE A2G2 | F6 |
e2 B3 G | c2 F4 | B3 A GF | GF GA B2 |
ed cB Ac | BA GA BG | B,G F3 E | E6 |]
G3 A Bc | dB AB G2 | e3 d ce | d6 |
c3 B Ac | BA G2 B,2 | CE A2 G2 | F6 |
E3 F GA | BG A2 B2 | c3 d ef | g4 fe |
dg fe dB | ce dc BG | A3 G FD | E6 :|]
W:
W:Bars one to eight are the original fragment.
W:Bars nine to sixteen are a guess at the second half of the A part.
W:Bars seventeen to thirtytwo are an invented B part using the resolutions from part A
W:W:Bars thirtythree to fortyeight following leeneia's suggestion to start in Relative Major key.

Has anyone else got their own solutions?