“…where the bar denotes the closure of the set and L is the linearization of the vector field at the origin. In this case, the normal form process introduces additional symmetries into the problem since every vector field is formally conjugated to a vector field with symmetry group S. Several works deal with vector fields that present symmetric geometric configurations, as we can see in [4,5,9,15,18]. In particular, in the same way as Elphick et al [14] and based on algebraic invariant theory, Baptistelli, Manoel and Zeli present a method ([4, Theorem 4.7]) to obtain normal forms of a reversible equivariant vector field that preserve the symmetries of the original vector field.…”