The analytic and formal normal form for the nilpotent singularity (Q1598387)

Two-dimensional analytic vector fields with nilpotent singularity are treated by the authors. They prove that the Bogdanov-Takens prenormal form is analytic. They give also the complete formal orbital classification in a certain special case. As a corollary, they show that such a system with a centre has a local analytic first integral if and only if it is analytically orbitally equivalent to a certain Hamiltonian system. The proofs are very technical and difficult. The analyticity is proved by estimating the coefficients of the Taylor series expansions.