Abstract. 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 f 1 , . . . , fm, defined around p, such that (df 1 ∧. . .∧dfm)(p) = 0 and d(df j (J( )))(p) = 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 .Introduction. Consider a (1, 1) tensor field J, defined on a real or complex mdimensional manifold M , whose Nijenhuis torsion vanishes. Suppose that for each point. In this paper we give a complete local classification of J on a dense open set that we call the regular open set. Moreover, near each regular point, i.e. each element of the regular open set, J is written with affine coefficients on a suitable coordinate system.To express the condition about functions f 1 , . . . , f m , stated above, in a simple computational way we introduce the invariant P J (see section 1). This invariant only depends on the 1-jet of J at each point, and P J (p) = 0 iff functions f 1 , . . . , f m as before exist. When J defines a G-structure, the first-order structure function being zero implies P J = 0 and N J = 0 (this last property is well known). Besides all points of M are regular; therefore this work generalizes the main result of [5]. On the other hand N J and P J both together can be considered as a generalization of the first-order structure function.This kind of tensor fields appear in a natural way in Differential Geometry. For example, on the base space of a bilagrangian fibration (see [1]) there exists a tensor field J, 1991 Mathematics Subject Classification: Primary 53C15; Secondary 58H05, 35N99. Key words and phrases: (1,1) tensor field, bihamiltonian structure. Supported by DGICYT under grant PB91-0412. The paper is in final form and no version of it will be published elsewhere.[449] 450 F. J. TURIEL with N J = 0, such that if (x 1 , . . . , x m ) are action coordinates then each dx j • J is closed; so P J = 0. From a wider viewpoint, when N J = 0, we can study the equation:i.e. the existence of conservation laws for J. Our classification shows that the existence, close to p, of m functionally independent solutions to equation (1) is equivalent to P J = 0 near p. Partial answers to the foregoing question may be found in [2], [6] and [7]. In [4], by using eigenvalues and eigenspaces, J. Grifone and M. Mehdi give an elegant necessary and sufficient condition for the existence of enough local solutions to equations (1) when J is real analytic. With the Grifone-Mehdi condition all points are regular and a calculation shows that it implies P J = 0. Therefore the Grifone-Mehdi result follows from ours.Finally, let us sketch the way for classifying J. As N J = 0 we can construct a bihamiltonian structure on T * M and from it a (1, 1) tensor field J * , prolongation of J to T * M (see [8]). The mai...