A general symmetry preserving reduction scheme and normal form for   dynamical systems with a compact symmetry group
  We present a generalized Lyapunov Schmidt reduction scheme for diffeomorphisms living on a finite dimensional real vector space V which transform under real one dimensional characters of an arbitrary compact group with linear action V. Moreover we prove a normal form theorem, such that the normal form still has the desirable transformation properties with respect to the character. 
Maria Cristina Ciocci, Johan Noldus