Metric indecomposability

If L is the region
H(,) = E
every invariant region, f(R) = R, has (R) = (L) or (R) = 0.

Metric indecomposability is equivalent to the non-existence of integrals of the motion except E.

Proof:
If the trajectory is decomposable: Set g(,) = 0 on one branch, g(,) = 1 on the other. g(,) is a constant for each of the branches. This proves that decomposability implies the existence of an integral of motion.
If g(,) is a constant and take different values: Choose c and use g(,) c and g(,) > c to define a decomposition. This proves that the existence of an integral of motion implies decomposability and completes the proof.