 Solution

A constant of motion is a quantity preserved along the evolution, possibly with explicit time dependence. For $C(\mathbf{x},\mathbf{v},t)$ in a gravitational potential $\Phi$, the preservation condition isFor free motion, $\mathbf{x}-\mathbf{v}t$ is a useful explicitly time-dependent example. Such initial-position labels should not be counted as stationary isolating integrals of a galactic orbit.