Nambu structures and integrable 1-forms

Abstract

Some years ago Moshé Flato pointed up that it could be interesting to develop the Nambu's idea to generalize Hamiltonian mechanic. An interesting new formalism in that direction was proposed by T. Takhtajan. His theory gave new perspectives concerning deformation quantization, and many authors have developed its mathematical features. The purpose of this paper is to show that this theory, at first designated to physic, gives a new point of view for the study of singularities of integrable 1-forms. Namely, we will prove that any integrable 1-form which vanishes at a point and has a non-zero linear part at this point is, up to multiplication by a non-vanishing function, the formal pull-back of a two dimensional 1-form. We also obtain a classification of quadratic integrable 1-forms.