  We consider the maximal supersymmetric pure Yang-Mills theories on six and
eight dimensional space. We determine, in a systematic way, all the possible
fractions of supersymmetry preserved by the BPS states and present the
corresponding `self-dual' BPS equations. In six dimensions the intrinsic one
has 1/4 supersymmetry, while in eight dimensions, 1/16, 2/16, ..., 6/16. We
apply our results to some explicit BPS configurations of finite or infinite
energy on commutative or noncommutative spaces.
