Well, assuming your calculations are correct (and that I understand your question), you need to find appropriate contours C1 and C2 such that:
t(t-1)-1ext (*)
is zero on dCi, i=1,2. So just choose your contours such that the endpoints lie at the zeros of (*), or a closed contour* that doesn't generate a trivial solution (i.e one that doesn't give yi=0). So at what points does (*) vanish in the (extended) complex plane? And is there a closed contour we could use that contains a simple pole (and hence generates a non-trivial solution)?
*This works because if Ci is closed dCi = Ø.