## Lie Symmetries for Implicit Planar Webs

J. Math. Sci. Univ. Tokyo
Vol. 29 (2022), No. 1, Page 115-148.

Hénaut, Alain
Lie Symmetries for Implicit Planar Webs

Abstract:
Let $F(x, y, y') = 0$ be an analytic or algebraic differential equation with $y'$ degree $d$. We deal with the qualitative study ofsuc h equation through the geometry ofthe planar $d$-web generated by the integral curves. Using meromorphic connection methods associated with the analytic class of $F$, Lie or infinitesimal symmetries of these configurations are studied for essentially $d \ge 3$ in the nonsingular case and from the viewpoint of their singularities. Maximal rank problems related to Abel’s addition theorem are also discussed. Basic examples are given from different domains including classic algebraic geometry and Frobenius 3-manifolds or WDVV-equations.

Keywords: Web geometry, Lie algebra of symmetries, abelian relations, Frobenius 3-manifolds or WDVV-equations.

Mathematics Subject Classification (2010): 14C21, 53A60, 32S65.