Question: Given a collection of point-particles, interacting through an attractive force $\sim \frac{1}{r^2}$.

Knowing only $m\_1a=\sum\_i \frac{Gm\_1m\_i}{r^2}$ and initial conditions we can deduce the motion of the system.

Consequently we can observe that three quantities remains constant
A) center of mass of the system
B) total energy
C) angular momentum

How can we derive these 3 facts directly from $m\_1a=\frac{Gm\_1m\_2}{r^2}$ ?

Are these quantities conserved for any attractive force $\sim\frac{1}{r^n}$ ?

Given any monotonically decreasing force for $r$ in $(0,\infty)$, which are the conserved quantities?

Answer: 
The total momentum is also conserved.

Noether's theorem relates conserved quantities to continuous symmetries of the system.

For the inverse square law, there's also a less known symmetry
$x \to e^\lambda x$, $t \to e^{2\lambda} t$.

But we shouldn't expect any more because of chaos; this isn't an integrable system.
There are many more conserved quantities in a potential N-body problem but not all of them are **additive** on particles. This fact is not related to the form of potentials at all.