 Solution

In the distinction relevant here, an integral of motion is a single-valued function $I(\mathbf{x},\mathbf{v})$ of instantaneous phase space, with no explicit time dependence, constant along the orbit. In an autonomous Hamiltonian system this is $\{I,H\}=0$, where the bracket is the Poisson bracket. An integral must be defined consistently on the phase space region under discussion; an arbitrary multivalued formula obtained by solving one trajectory does not supply a global integral. Some conventions call explicitly time-dependent conserved functions integrals too; the stationary interpretation is needed for the following counts.