Normal form theory for reversible equivariant vector fields

We give an alternative method to obtain normal forms of reversible equivariant vector fields. We adapt the classical method using tools from invariant theory to establish formulae that take symmetries into account as a starting point. Normal forms of two classes of non-resonant and resonant cases are presented, both under a Z_2-action with linearization having a 2-dimensional nilpotent part and a semisimple part with purely imaginary eigenvalues.