As it is well known, the global structure of the Einstein equations for
general relativity in the context of the initial value problem, is a difficult
and intricate mathematical problem. Therefore, any additional structure in
their formulation is useful as a tool for studying the global behaviour of the
initial value problem of the Einstein equations. In our previous works, we have
used the additional structure provided by the dimensional reduction to $2+1$
dimensional Einstein-wave map system. In this work, we shall focus on yet
another structure in the Einstein equations for the U(1) symmetric spacetimes,
namely the analogy with the Yang-Mills theory and reconcile with the
dimensionally reduced field equations.

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