Generalization of Jacobi's theorem on the last multiplier

To satisfy the conditions of Jacobi's theorem on the last multiplier, it is needed the existence of invariant measure and the presence of a sufficient number of independent first integrals. In this case, the system is locally integrated in quadratures. There are known the examples of systems in which it turned out that for the possibility of integration in quadratures it is sufficient the existence of partial first integrals. In this case, integration in quadratures occurs at the levels of partial first integrals.

In this paper, Jacobi's theorem on the last multiplier is extended to the general situation, when among the first integrals there are partial integrals.