  A major problem in discussing heterotic string models is the stabilisation of
the many vector bundle moduli via the superpotential generated by world-sheet
instantons. In arXiv:1110.6315 we have discussed the method to make a discrete
twist in a large and much discussed class of vector bundles such that the
generation number gets new contributions (which can be tuned suitably) and at
the same time the space of bundle moduli of the new, twisted bundle is a proper
subspace (where the 'new', non-generic twist class exists) of the original
bundle moduli space; one thus gets a model, closely related to the original
model one started with, but with enhanced flexibility in the generation number
and where on the other hand the number of bundle moduli is {\em somewhat}
reduced. Whereas in the previous paper the emphasis was on examples for the new
flexibility in the generation number we here classify and describe explicitly
the twists and give the precise reduction formula (for the number of moduli)
for SU(5) bundles leading to an SU(5) GUT group in four dimensions. Finally we
give various examples where the bundle moduli space is reduced {\em
completely}: the superpotential for such {\em rigid} bundles becomes a function
of the complex structure moduli alone (besides the exponential Kahler moduli
contribution).
