 Solution

The two continuous, time-independent symmetries are time translation and a common rotation of both angles. Their Noether charges are respectively the total energy and vertical angular momentum:For the second symmetry, $\theta \mapsto \theta +ϵ$ and $\phi \mapsto \phi +ϵ$, so the Noether theorem gives the sum of their conjugate momenta. Independent rotations are not symmetries of the spring coupling.

There are also discrete symmetries: simultaneous reflection $(\theta ,\phi )\mapsto (-\theta ,-\phi )$, and time reversal in classical mechanics, which reverses both angular velocities. These leave the dynamics invariant but provide no additional infinitesimal Noether charges. Combined with common rotations, the reflections give the full planar orthogonal symmetry of the angular geometry. If $I_1=I_2$, interchange of the two angular coordinates is an additional discrete symmetry of the reduced dynamics.

There is a further useful time-dependent symmetry, up to a total derivative, on a continuous lift of the angles: adding the same constant angular velocity, $(\theta ,\phi )\mapsto (\theta +vt,\phi +vt)$. It changes $L$ byIts conserved quantity is $\boxed{I_1\theta +I_2\phi -tJ}$. It is the integration constant for free common rotation, rather than a new globally single-valued angle charge on the configuration torus. The globally defined continuous charges are $E$ and $J$ above.