Completely integrable system with Jacobi multipliers and its KAM stability

In the present paper, we develop a general theory for the reduction of systems with Jacobi multiplier and volume-preserving symmetries on $ n $-manifolds ($ n\geq2 $). If the system has $ r $ commuting volume-preserving symmetries, we can reduce it to another system with a Jacobi multiplier, where the solutions of this reduced system are determined by the solutions of its first $ n-r $ equations. Specifically, if the system has $ n-2 $ such symmetries, then the solutions of the transformed equations are determined by the solutions of its first two equations, and these first two equations constitute a one-degree-of-freedom Hamiltonian system. We also use geometric methods to provide a local way to find the complete first integrals of the system. Additionally, we provide examples illustrating that using our method to find first integrals is equivalent to solving certain partial differential equations. Finally, we used a KAM-type theorem to investigate the stability of the solutions of this system.