Does the Lagrangian being invariant under substitution of variables imply a conserved quantity?

Question

Consider the following Lagrangian:

$$ \mathcal{L} = \frac{Ma^2\dot\theta^2}{6} +\frac{1}{2}ma^2\left(4\dot\theta^2 + \dot\phi^2 + 4\dot\theta\dot\phi\cos(\theta - \phi) \right) - \frac{a^2k}{2}\left( 21 - 16\sin\theta - 8\sin\phi + 4\cos(\theta - \phi) \right) - Mga\sin\theta - mga\left(2\sin\theta + \sin\phi \right) $$ and the substitution $(\theta, \phi) \rightarrow (\pi - \theta, \pi - \phi)$. The $\mathcal{L}$ is invariant under this change of variables. I was asked to tell if there is a conserved quantity associated to this transformation. I tried applying Noether's theorem, but in this case the transformation doesn't depend on any parameter, so it wouldn't make sense, I guess? I still think this is the right path, maybe I just didn't understand the theorem properly. I even tried inspecting the equations of motion of this system, with no success.

Answer 1

The symmetry is the inversion at the origin of the unit 2-sphere, that is a reflexion at the equator plane plus a rotation by $\pm \pi$. The conserved quantity is the parity of the eigenstates with equal eigenvalues (assuming the constants yield a positive metrics on the tangent space of the sphere).