Engel theorem through singularities

TL;DR

This paper establishes a singular version of the Engel theorem, providing a normal form for holomorphic singular Engel systems and analyzing their singular sets, with applications to integral curves and global decomposability.

Contribution

It introduces a singular version of the Engel theorem and a normal form for holomorphic singular Engel systems under certain conditions.

Findings

Existence of integral analytic curves through singular points.

Singular set of globally decomposable Engel systems has atypical codimension.

Normal form theorem for germs of holomorphic singular Engel systems.

Abstract

We prove a singular version of the Engel theorem. We prove a normal form theorem for germs of holomorphic singular Engel systems with good conditions on its singular set. As an application, we prove that there exists an integral analytic curve passing through the singular points of the system. Also, we prove that a globally decomposable Engel system on a four dimensional projective space has singular set with atypical codimension.