Noether's Inverse Theorem says that there must be a symmetry $\delta q^{i}$ given a conserved quantity $C$.
$$\delta q^{i}=\epsilon W^{ij}\frac{\partial C}{\partial q^{j}},$$
where $\epsilon$ is an arbitrary constant and $W^{ij}$ is the Hessian of the Lagrangian.
Basically, one can guess some $C$ to obtain $\delta q^{i}$ from this equation, but usually one knows from the Lagrangian which $C$ is suitable for doing this.
My question is: Is there a way of knowing what is not a conserved quantity for the system? Or a way to make this equation fail given a $C$?
In particular, I am working with the Lagrangian:
$$L=\frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^2)-V(x^{2}+y^{2}+z^{2})-\frac{m}{2}[(\dot{x}y-\dot{y}x)\omega^{2}+\omega^{2}(x^{2}+y^{2})]$$
where $V$ is a derivable function.
I found that the energy $E$ gives a specific symmetry. But, I am afraid of inserting the angular momentum and find that there is another symmetry, and actually afraid that anything that I would insert in the equation would give more symmetries... I think there should be something that tells me that some of those are actually not symmetries.
