The constant of motion of a free fall body

Main Question or Discussion Point

1.what are the the constants of motion of a free fall body(treat this as a one-dimension motion,so there are two [independent] constants of motion )?

2.are they initial velocity and initial displacement？

3.and i know the total energy(T+V) is one,but they are not independent with the initial velocity and initial displacement, isn't it?

4.so i means can we just treat the initial velocity and initial displacement as constants of motion ?

5.if we can,do these constant of motin (initial velocity and initial displacement) correspond to some symmetry of the system (free fall body)?

thanks

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