Integral of a vector field

It should be recalled that a smooth function / on a manifold M is called an integral of a vector field t if it is constant along the field u, that is, v(f) = 0. This is equivalent to the fact that the function / is constant on all integral trajectories of the field v. It is well known that the presence of an integral in a vector field v(x) = (vi(x),.. which corresponds to a system of ordinary differential... 

Thus, the search for a maximal possible number of independent integrals of a vector field is an important task. For Hamiltonian systems, the search for integrals is sometimes facilitated by specific properties of these systems. 

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