Singularities of two-dimensional Nijenhuis operators

A Nijenhuis operator L is a (1, 1)-tensor field on a smooth manifold M with vanishing Nijenhuis torsion NL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal N_L}}$$\end{document}. At each point x∈M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in M$$\end{document}, the algebraic type of L(x) is characterized by its Jordan normal form. In this paper we study singularities of a two-dimensional Nijenhuis operator in the case when its trace trL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{tr}\,}}L$$\end{document} has a non-zero differential at the singular point. A description of such singularities reduces to studying the smoothness of some function which is a fraction depending on partial derivatives of the determinant of L. We completely describe singularities for some special classes of functions. We also obtained interesting examples of Nijenhuis operators and their singularities.