The Einstein-Yang-Mills equations are the source of many interesting solutions within general relativity, including families of particle-like and black hole solutions, and critical phenomena of more than one type. These solutions, discovered in the last thirty years, all assume a restricted form for the Yang-Mills gauge potential known as the magnetic ansatz. In this thesis we relax that assumption and investigate the most general solutions of the Einstein-Yang-Mills system assuming spherically symmetry, a Yang-Mills gauge group of SU(2), and zero cosmological constant. We proceed primarily by numerically integrating the equations and find new static solutions, for both regular and black hole boundary conditions, which are not asymptotically flat, and attempt to classify the possible static behaviours. We develop a code to solve the dynamic equations that uses a novel adaptive mesh refinement algorithm making full use of double-null coordinates. We find that the type II critical behaviour in the general case exhibits non-universal critical solutions, in contrast to the magnetic case and to all previously observed type II critical behaviour.
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