  Poisson brackets admit infinitesimal symmetries which are encoded using
oriented graphs; this construction is due to Kontsevich (1996). We formulate
several open problems about combinatorial and topological properties of the
graphs involved, about integrability and analytic properties of such symmetry
flows (in particular, for known classes of Poisson brackets), and about
cohomological, differential geometric, and quantum aspects of the theory.
  Keywords: Poisson bracket, Jacobi identity, infinitesimal symmetry, affine
manifold, Poisson cohomology, graph complex.
