Normal forms and linearization of resonant vector fields with multiple eigenvalues

We discuss the linearization and normal forms of resonant vector fields X(x)=Ax+a(x), where A has one double or triple eigenvalue or a pair of double eigenvalues: we present a simple way of identifying the resonant monomials that have to appear in its normal form, and also effective conditions on the nonlinearity a(x) for the resonant vector field to be linearizable.