The authors classify all such pre-foliations up to automorphism by explicit one-forms parametrized by Gauss-map ramification data, completing the cases for tangent sheaf degrees 3 and 4.
Bedrouni, Pre-foliations of co-degree one on ^2 with a flat Legendre transform, arXiv preprint arXiv:2309.xxxx (2023)
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Homogeneous pre-foliations of co-degree one and degree four on the projective plane
The authors classify all such pre-foliations up to automorphism by explicit one-forms parametrized by Gauss-map ramification data, completing the cases for tangent sheaf degrees 3 and 4.