  We are going to analyze through a first order perturbative formulation the
local loss of symmetry when a source of non-Abelian Yang-Mills and
gravitational fields interacts with an external agent that perturbes the
original geometry associated to the source. Then, as the symmetry in Abelian
and non-Abelian field structures in four-dimensional Lorentzian spacetimes is
displayed through the existence of local orthogonal planes of symmetry that we
previously called blades one and two, the loss of symmetry will be manifested
by the tilting of these planes under the influence of the external agent. It
was found already that there is an algorithm to block diagonalize the
Yang-Mills field strength isospace projections in a local gauge invariant way.
Independently, it was also found an algorithm to diagonalize the Yang-Mills
stress-energy tensor in a gauge invariant way. Using these results and
perturbative analysis from a previous manuscript dealing with the Abelian case,
we are going to demonstrate how to develop an algorithm for constructing local
energy-momentum conserved currents inside both local orthogonal planes. As the
interaction proceeds, the planes are going to tilt perturbatively, and in this
strict sense the original local symmetries will be lost. But we will prove that
the new blades at the same point will correspond after the tilting generated by
perturbation, to new symmetries, with associated new local currents, both on
each new local planes of symmetry. Old symmetries will be broken, new will
arise. There will be a local symmetry evolution in the non-Abelian case as
well.
