## On Lins Neto’s examples of algebraic foliations. (Sur les exemples de Lins Neto de feuilletages algébriques.)(French. Abridged English version)Zbl 1004.37029

Let $$X_0$$ and $$X_\infty$$ be holomorphic vector fields on $$\mathbb{C}^3$$ defined by $X_0=(-3z_1^2+ z^2_2+2z_1z_3) \partial/\partial z_1+2z_2(-3z_1+2z_3) \partial/ \partial z_2+2z_3 (3z_1-z_3)\partial/ \partial z_3$ and $X_\infty= 2z_2(-z_1+z_3) \partial/ \partial z_1+(3z_1^2- z^2_2)\partial/ \partial z_2+2z_3z_2 \partial/ \partial z_3,$ where $$(z_1,z_2,z_3)$$ denotes the coordinates on $$\mathbb{C}^3$$. Set $$X_\alpha= X_0+\alpha X_\infty$$ for $$\alpha\in \mathbb{C}$$. Let $$\Lambda$$ be the lattice in $$\mathbb{C}$$, generated by 1 and $$\omega$$, where $$\omega$$ is a primitive 6-th root of unity. The author proves, among other things, that $$X_\alpha$$ is completely integrable if and only if $$(\alpha-1)/2= \lambda_1/ \lambda_2$$ for some $$\lambda_1$$, $$\lambda_2 \in\Lambda$$. The author asserts that the family $$\{X_\alpha\}$$ gives a counterexample to the conjecture of Painlevé, which says that every polynomial vector field on $$\mathbb{C}^n$$ with global meromorphic solutions is completely integrable.

### MSC:

 37F75 Dynamical aspects of holomorphic foliations and vector fields 32S65 Singularities of holomorphic vector fields and foliations 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 57R30 Foliations in differential topology; geometric theory

### Keywords:

complete integrability; holomorphic vector fields
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### References:

 [1] Ercolani, N.; Siggia, E. D., Painlevé property and integrability, (What is Integrability? (1991), Springer: Springer Berlin), 63-72 · Zbl 0733.34008 [2] Ghys, É.; Rebelo, J.-C., Singularités des flots holomorphes, II, Ann. Inst. Fourier (Grenoble), 47, 4, 1117-1174 (1997), Erratum op. cit. 50 (3) (2000) 1019-1020 · Zbl 0938.32019 [3] A. Lins Neto, Some examples for Poincaré and Painlevé problems, Prépublication, 2000; A. Lins Neto, Some examples for Poincaré and Painlevé problems, Prépublication, 2000
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