A separability theorem for dynamical systems admitting alternative Lagrangian descriptions

By generalising earlier results, we prove that whenever a Lagrangian dynamical system can be associated with a (1,1)-type tensor field which is left invariant by the dynamics, satisfies the Nijenhuis condition and is compatible with the tangent bundle structure, the system necessarily decomposes into a collection of lower-dimensional mutually non-interacting Lagrangian subsystems.