  A field of endomorphisms $R$ is called a Nijenhuis operator if its Nijenhuis
torsion vanishes. In this work we study a specific kind of singular points of
$R$ called points of scalar type. We show that the tangent space at such points
possesses a natural structure of a left-symmetric algebra (also known as
pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to
linearization of Poisson structures, we state the linearisation problem for
Nijenhuis operators and give an answer in terms of non-degenerate
left-symmetric algebras. In particular, in dimension 2, we give classification
of non-degenerate left-symmetric algebras for the smooth category and, with
some small gaps, for the analytic one. These two cases, analytic and smooth,
differ. We also obtain a complete classification of two-dimensional real
left-symmetric algebras, which may be an interesting result on its own. This
work is the second part of a series of papers on Nijenhuis Geometry started
with arXiv:1903.04603
