Classification of (1,1) tensor fields and bihamiltonian structures

Consider a (1,1) tensor field J, defined on a real or complex m-dimensional manifold M, whose Nijenhuis torsion vanishes. Suppose that for each point p ∈ M there exist functions ${\displaystyle f_1,...,f_m}$, defined around p, such that ${\displaystyle (d{f}_1\wedge ...\wedge d{f}_m)\left(p\right)\ne 0}$ and ${\displaystyle d\left(d{f}_j\left(J\left(\right)\right)\right)\left(p\right)=0}$, j = 1,...,m. Then there exists a dense open set such that we can find coordinates, around each of its points, on which J is written with affine coefficients. This result is obtained by associating to J a bihamiltonian structure on T*M.