  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.
