Source-free Yang-Mills theories

Using techniques developed in recent years in general relativity we study the general GL($R,n$) and GL($C,n$) gauge theories with no external sources. In particular, after casting the field equations into the spin-coefficient formalism and adapting natural (to the formalism) gauge conditions we show how most of the equations can be integrated in the case of real asymptotically "flat" solutions. In the self-dual (or anti-self-dual) case the field equations are reduced to a single nonlinear wave equation for a Hertz-type potential. As a final point we show the relationship between this potential and our version of the Atiyah-Ward method of finding self-dual fields.