Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators

A field of endomorphisms R is called a Nijenhuis operator if its torsion vanishes. In this work we study specific kind singular points scalar type. We show that the tangent space at such possesses natural structure left-symmetric algebra (also known as pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to linearization Poisson structures, state linearisation problem for operators and give an answer in terms non-degenerate algebras. particular, dimension 2, classification algebras smooth category and, with some small gaps, analytic one. These two cases, smooth, differ. also obtain complete two-dimensional real algebras, which may be interesting result on own.