ИСТИНА

We discuss an important class of Lie-Reinhart algebras, the so-called graded polynomial Lie algebras, introduced by Viktor Buchstaber to study some integrable systems. This class of Lie algebras, in particular, contains the Lie algebras of polynomial vector fields. Polynomial Lie algebras also arise naturally as characteristic Lie algebras of certain hyperbolic partial differential equations, for example, the Klein Gordon equation. We discuss the relationship between the growth of the characteristic Lie algebra and the integrability of the corresponding PDE (the existence of Darboux integrals and higher symmetries).