As a first approach to the study of classical solutions for gauge theories in dimensions higher than four we study linear relations among the fields which ensure the equation of motion as a consequence of the Bianchi identities. We show how these relations can be obtained as integrability conditions on linear covariant first order equations. Using spherical symmetry we exhibit specific solutions of. these relations. We also hint at an algebraic approach of the ADHM type to obtain more general solutions at least in principle.