Recently it has been pointed out that the so-called Faddeev-Niemi equations that describe the Yang-Mills equations of motion in terms of a decomposed gauge field, can have solutions that obey the standard Yang-Mills equations with a source term. Here we argue that the source term is covariantly constant. Furthermore, we find that there are solutions of the Yang-Mills equation with a covariantly constant source term that are not solutions to the Faddeev-Niemi equations. We also present a general class of gauge field configurations that obey the Faddeev-Niemi equation but do not solve the Yang-Mills equation. We propose that these configurations might have physical relevance in a strongly coupled phase, where spin-charge separation takes place and the Yang-Mills theory cannot be described in terms of a Landau liquid of asymptotically free gluons.

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