We develop a constructive approach to the problem of polynomial first integrals for linear vector fields. As an application, we obtain a new proof of the theorem of Wietzenbock about finiteness of the number of generators of the ring of constants of a linear derivation in the polynomial ring. Moreover, we propose an alternative approach to the analyticity property of the normal form reduction of a germ of vector field with nilpotent linear part in a case considered by Stolovich and Verstringe.