Liouville's theorem states that phase space volume is conserved over time with respect to the dynamical system generated by the Hamiltonian and Hamilton's equations.

However, any given point in phase space will evolve within a submanifold characterized by certain values of the conserved quantities (energy, momentum,...).

It's not obvious to me that the "phase volume" within this submanifold is also conserved over time, since it is a volume of lower dimension than that of tbe phase space.

Is there a result here that you could point me to?