Dynamics of the system in the presence of invariant relationships

The possibility of the existence of an invariant measure with smooth density is discussed in two cases related to invariant sets – at the levels of particular integrals and at the joint invariant level of two or more functions. Conditions are investigated when the invariant sets represent a two-dimensional torus on which an invariant measure with smooth density is defined, which means that Kolmogorov’s theorem is applicable, and the motion after the appropriate coordinate replacement is conditionally periodic.