Symmetric matrices and maximal Nijenhuis pencils

Abstract

A Nijenhuis pencil is a linear subspace of the space of $(1,1)$ tensor field which consists of Nijenhuis operators. The problem of the description of maximal (by inclusion) Nijenhuis pencils containing a subpencil of dimension $n(n+1)/2$ such that the operators in it are — in some system of coordinates — constant symmetric matrices, is solved. Two such pencils turn out to exist, both of which arise in a natural way in applications, for example, in the theory of infinite-dimensional integrable systems.