In our classical mechanics class, professor said that Kepler's problem is a kind of Integrable System such that the number of conserved quantities would be equal to the number of degrees of freedom. In Kepler's problem, we know that the number of degrees of freedom would be $3+3=6$ since there are $2$ objects connecting by gravity. Then Professor counted the number of conserved quantities for us, 3 for the components of angular momentum, and the remaining 3 are for the components of Laplace-Runge-Lenz vector. However, I wonder why he didn't count the mechanical energy as a conserved quantity? Can someone tell me why?
He could (and should) count the mechanical energy as a conserved quantity. He also didn't mention that the total momentum of the 2 objects is 3 integrals of motion. However, not all quantities are independent.
If we focus on the relative motion in the center-of-mass system, then there are $n=3$ DOF. The phase space has $2n=6$ dimensions. The mechanical energy, the angular momentum and the Laplace-Runge-Lenz vector are 1+3+3=7 integrals of motion. However, there can only be $2n-1=5$ independent integrals of motion in a superintegrable system. In fact one may show that the relative Kepler problem is maximally superintegrable.