A system having N constants of motion means having N fundamental, mathematically derived quantities that are constant for the motion. $F = m dv/dt$ is physically derived from experiment. You can create the mathematically derived quantity work by forming the scalar product of both sides with a differential displacement:
$F\cdot dl = m dv/dt \cdot dl = m v \cdot dv$
Integrating over conservative forces, you get a constant of motion, kinetic energy + potential energy = constant. You can create another mathematically derived quantity impulse by multiplying both sides by dt:
$F dt = m dv/dt dt = m dv$
Integrating over a system where all the forces are internal and action = reaction, you get another constant of motion, total momentum = constant.
Look in any standard text book for the details of the above derivations, and Noether’s theorem which is another way of deriving constants of motion from physically derived quantities. The important point is they’re fundamental, mathematically derived constants of motion from physically derived quantities.