Euler-Lagrange equations and symmetries

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I'm hoping this is a really simple question, but I can't seem to find a definitive answer anywhere!

If the action is invariant under some symmetry transformation, do the equations of motion need to be invariant as well?

The trouble I'm having relates to SU(N) yang-mills theories where I'm finding that even if the action is invariant under the SU(N) local gauge transformation, the equations of motion are not. I believe this is related to the fact that the lagrangian is not invariant, but changes by a total derivative term. However I'm not sure if I'm doing something wrong or this is actually possible...

*edit* also, if someone could explain the physical significance of this that would be amazing, because I am completely confused. For example in the SU(N) theory, the right side of the EOM (which is just the source term) is "rotated" after the transformation. i.e. [itex][D_\nu,F^{\mu\nu}]=J^\mu \rightarrow [D_\nu,F^{\mu\nu}]=U^\dagger J^\mu U[/itex]
 

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