Nijenhuis geometry III: gl-regular Nijenhuis operators

We study Nijenhuis operators, that is, (1,1)-tensors with vanishing torsion under the additional assumption they are gl-regular, i.e., every eigenvalue has geometric multiplicity one. prove existence of a coordinate system in which operator takes first or second companion form, and give local describtion such operators. apply this description to singular points. In particular, we obtain their normal forms dimension two discover topological restrictions for gl-regular operators on closed surfaces. This paper is an important step research programme suggested arXiv:1903.04603 arXiv:1903.06411.