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There are no magnetically charged particle-like solutions of the Einstein Yang-Mills equations for Abelian models

We prove that there are no magnetically charged particle-like solutions for Abelian models in Einstein Yang-Mills, but for non-Abelian models the possibility remains open. An analysis of the Lie algebraic structure of the Yang-Mills fields is essential to our results. In one key step of our analysis we use invariant polynomials to determine which orbits of the gauge group contain the possible asymptotic Yang-Mills field configurations. Together with a new horizontal/vertical space decomposition of the Yang-Mills fields this enables us to overcome some obstacles and complete a dynamical system existence theorem for asymptotic solutions with nonzero total magnetic charge. We then prove that these solutions cannot be extended globally for Abelian models and begin an investigation of the details for non-Abelian models.