  The instanton equations on vector bundles over Calabi-Yau and hyper-K\"ahler
cones can be reduced to matrix equations resembling Nahm's equations. We
complement the discussion of Hermitian Yang-Mills (HYM) equations on Calabi-Yau
cones, based on regular semi-simple elements, by a new set of (singular)
boundary conditions which have a known instanton solution in one direction.
This approach extends the classic results of Kronheimer by probing a relation
between generalised Nahm's equations and nilpotent pairs/tuples. Moreover, we
consider quaternionic instantons on hyper-K\"ahler cones over generic
3-Sasakian manifolds and study the HYM moduli spaces arising in this set-up,
using the fact that their analysis can be traced back to the intersection of
three Hermitian Yang-Mills conditions.
