Unique normal forms for the Takens-Bogdanov singularity in a special case (Q2721628)

The authors deal with a planar vector field in equilibrium at the origin with nonzero nilpotent linear part. They consider the special case: NEWLINE\[NEWLINEF(x,y)= y\partial_x+ (x^2+\beta_1 xy\partial_y)+ \sum_{j=2}^\infty (\alpha_j x^{j+1}+ \beta_j x^jy) dy. \tag{1}NEWLINE\]NEWLINE Their goal is to present a further reduction of (1) and to give a unique normal form for the special case of an open problem for the Takens-Bogdanov singularity.