Invariants of group actions, dim/deg duality and normal forms of vector fields

In 1932 R. Wetzenbock proved that the ring of polynomial invariants of a linear vector field in C^k is finitely generated. The proof eventually boils down to its most difficult case, when the vector field is X=x_2\partial_1+...+x_k \partial_{k-1}, i.e., is defined by a matrix being a nilpotent Jordan cell. With X is associated another nilpotent linear vector field Y, such that they together generate a representation of the Lie algebra sl(2,C). The first integrals of Y appear in a multidimensional generalization of so-called Takens normal form for germs V=X+....

Together with E. Stróżyna we develope a contructive approach to the problem of invariants of X. In particular, we obtained an alternative proof of the analyticity of the normal form for V in a special case (considered by L. Stolovich and F. Verstringe) when that form equals X+f(x)Y for a series f(x) being a simultaneous first integral of X and of Y.