If a system has $N$ degrees of freedom (DOF) and therefore $N$ independent1
conserved quantities integrals of motion, can continuous symmetries with a total of $N$ parameters be found that deliver these conserved quantities by means of Noether's theorem? I think this is not exactly the opposite of Noether's theorem since I don't ask whether for each conserved quantity a symmetry can be retrieved, I ask about a connection between the whole set of conserved quantities and symmetries.
1) or $2N-1$, or $N$, depending on definition and details that are irrelevant here. But let me expand on it anyway... I consider the number of DOFs equal to the number of initial conditions required to fully describe a system in Classical Mechanics. That means, velocities (or momenta) are considered individual DOFs, and not that each pair of coordinate + velocity make up only one DOF. Time is no DOF however, it is a parameter. Please discuss this in this question if you disagree.