Page 2 (506), line 18 of http://www-personal.umich.edu/~orosz/articles/NonlinScipublished.pdf
says that "The presence of translational symmetry in the nonlinear equations gives rise to a relevant zero eigenvalue in the linearized system at any of the trivial solutions".
It looks like a general statement, but I don't see why (is it trivial?). Where can I find a precise statement about this and on what conditions does that hold? (I'd like to know the proof too.) Thank you.