# American Institute of Mathematical Sciences

July  2009, 25(2): 481-493. doi: 10.3934/dcds.2009.25.481

## On the integrability of holomorphic vector fields

 1 Departamento de Matemática, Universidade Federal do Espírito Santo, Av. Fernando Ferrari 514, 29075-910, Vitória, ES, Brazil 2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21.945-970, Rio de Janeiro, RJ, Brazil

Received  February 2008 Revised  January 2009 Published  June 2009

We determine topological and algebraic conditions for a germ of holomorphic foliation $\mathcal{F}_X$ induced by a generic vector field $X$ on $(\mathbb{C}^{3},0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F$ : $(\mathbb{C}^{3},0)\rightarrow(\mathbb{C}^{2},0)$ such that the leaves of $\mathcal{F}_X$ are contained in the level curves of $F$.
Citation: Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
 [1] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [2] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

2019 Impact Factor: 1.338