Integrability of vector fields versus inverse Jacobian multipliers and normalizers

Abstract

In this paper we provide characterization of integrablity of a system of vector fields via inverse Jacobian multipliers (matrix) and normalizers of smooth (or holomorphic) vector fields. These results improve and extend some well known ones, including the classical holomorphic Frobenius integrability theorem. Here we obtain the exact expression of an integrable system of vector fields acting on a smooth function via their known common first integrals. Moreover we characterize the relations between the integrability and the existence of normalizers for a system of vector fields. In the case of integrability of a system of vector fields we not only prove the existence of normalizers but also provide their exact expressions.

Analytic partial-integrability of a symmetric Hopf-zero degeneracy

“…The inverse Jacobi multiplier is a useful tool in the study of vector fields. So, for example, the existence of a class of inverse Jacobi multipliers (or inverse integrating factors) has been used for the study of the Hopf bifurcation and centre problem, see [9–11, 16, 24], and for the integrability problem in general, see [2, 8, 21].…”