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**Analytic families of differential equations on two-dimensional varieties. Resolution of singularities, and singular perturbations.**
*(English.
Russian original)*
Zbl 0823.32012

Russ. Acad. Sci., Dokl., Math. 48, No. 1, 162-167 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 331, No. 5, 559-562 (1993).

The author proves that the singularities of all equations of the family of direction fields can be simultaneously decomposed into elementary ones. The proof of his main theorem is based on the proof of Bendixson-Seidenberg theorem which was proposed by Van den Essen and reworked by him for the algebro-geometric construction, without mensioning terms from the theory of differential equations, of unfolding of the base in a family. The key concerns to applications of unfoldings of the basis and lies in the so-called section theorem.

Reviewer: J.Kajiwara (Fukuoka)

### MSC:

32G34 | Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) |

32S45 | Modifications; resolution of singularities (complex-analytic aspects) |