Moscow--Petersburg seminar on low-dimensional mathematics SPb, PDMI, 26.04.2002, 16:00--17:45, room 311 B.Kruglikov "Mayer brackets and solvablity of scalar PDEs". In the talk we review some joint results with V.Lychagin. Obstructions to integrability of PDE systems belong to the second Spencer $\delta$-cohomology group and are called Weyl tensors. We study scalar equations with minimal overdetermination, i.e. given by two (non-linear) relations. In this case all Spencer cohomologies can be calculated and there is only one non-trivial obstruction, which we identify with the (higher) Mayer bracket. This produces a simple integrability criterion. The proof is based on an importent reduction theorem, which allows to study other (not necessary scalar) PDEs, satisfying the so-called Cohen-Macaulay conition, by a reduction to non-characteristic systems.