  We develop techniques for computing the integer valued SU(3) Casson
invariant. Our method involves resolving the singularities in the flat moduli
space using a twisting perturbation and analyzing its effect on the topology of
the perturbed flat moduli space. These techniques, together with Bott-Morse
theory and the splitting principle for spectral flow, are applied to calculate
the invariant for all Brieskorn homology spheres.
