On the integration of reduced holomorphic differential equations in dimension two. (Sur l’intégration des équations différentielles holomorphes réduites en dimension deux.) (French) Zbl 0969.34074

The aim of this paper is to understand what should be a (pseudo) Galois group for a nonlinear differential equation of order one. More precisely, it examines the case when the equation have “computable” solutions and hence when the Galois group is expected to be resoluble. Let \( \omega=0 \) be a germ of a holomorphic differential equation near \( (0,0) \) in \( \mathbb C^{2}\). The problem is to find if it has Nilsson first integrals (namely multiform functions with moderate growth along an analytic hypersurface) or, more generally, Liouvillian first integrals (namely in an extension of the field of meromorphic functions obtained in finitely many steps by taking a finite algebraic extension or adding a primitive or the exponential of a primitive). Using desingularisation, one can assume \( \omega \) to be under reduced form. Moreover, when not linearisable (in the “resonant” case) the differential equation can be put under an analytical (resp. formal) normal form.
The authors show that \( \omega \) has a Nilsson first integral only when it is analytically conjugated to its formal normal form. The same is not true in the Liouvillian case and the authors list the differential equations that cannot be analytically reduced to their formal normal form but have Liouvillian first integrals.
This paper is mainly self contained and, by giving explicit computations and examples, is very pleasant to read.


34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
12H20 Abstract differential equations
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
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