We confirm a conjecture of Bernstein–Lunts which predicts that the characteristic variety of a generic polynomial vector field has no homogeneous involutive subvarieties besides the zero section and subvarieties of fibers over singular points.
It is well-known that the existence of transversally intersecting separatrices of hyperbolic periodic solutions leads, in a typical situation, to complicated and irregular dynamics. Therefore, in the case of a two-dimensional mapping or a three-dimensional flow, with this transversality property, there is no non-trivial analytic or meromorphic first integral, i.e., a function constant along each trajectory of the system under consideration. Additional robust conditions are obtained and discussed...