Quasi-homogeneous normal forms for null linear part

In this paper, we analyze simplified normal forms for equilibria of vector fields with zero linear part, assuming the presence of a symmetry. We put in correspondence the vector field with zero linear part with a Takens-Bogdanov singularity with an invariant Axis, which we study by using expansions in quasi-homogeneous components of a fixed type. We perform both, near-identity transformations as well as reparametrizations in the time to determine the structure of the normal form. A degenerate case, in which a normal form coefficient vanishes, is also included.