Integrability Of Systems Of Vector Fields Versus Normalizers And Normal Forms

In this dissertation we first characterize the integrability of systems of s-mooth (or holomorphic) vector fields via inverse Jacobian multipliers (or inverse Jacobian multiplier matrices) and their common normalizers. This results im-prove and generalize some known relative results, for example the classical holo-morphic Frobenius integrability theorem, and the results on the relation between integrable vector field and Jacobian multiplier by Berrone and Giacomini [Rend. Circ. Mat. Palermo (Serie Ⅱ), LⅡ (2003),77-130]. Next, with the help of the common first integrals of systems of integrable vector fields we not only prove the existence of normalizers of the vector fields and also prove their exact expressions. There results are the generalization of those in Peralta-Salas [J.Differential Equa-tions 244(2008),1287-1303] and in Prince [J. Differential Equations 246(2009), 3750-3753] on the relation between integrability and the existence of normalizers for a singer vector field to a system of vector fields. Finally we provide an equiv-alent characterization on the existence of first integrals of analytic differential system near a non-hyperbolic singularity.