= Solution
A <continuous symmetry> is represented by a unitary family $U(\epsilon)=e^{-i\epsilon G/\hbar}$ with a self-adjoint <conserved generator of a continuous quantum symmetry> $G$. If $UHU^\dagger=H$ for all $\epsilon$, differentiating gives $[H,G]=0$. Thus, for no explicit time dependence,
$$
\boxed{\frac{d\langle G\rangle}{dt}=\frac{i}{\hbar}\langle[H,G]\rangle=0}.
$$
<translation> invariance conserves the corresponding <momentum> component; rotational invariance about one axis conserves that angular-momentum component. A <central potential> conserves all components and $L^2$, but the components need not be simultaneously measurable because $[L_x,L_y]=i\hbar L_z$. Time-translation invariance of a closed system gives a time-independent <Hamiltonian> and <energy> conservation.
A <discrete symmetry> need not have a continuously varying generator. For <parity>, $[H,P]=0$ lets stationary states be chosen even or odd, and preserves <parity> sectors. <symmetry> does not force every state to be an eigenstate of its conserved <observable>. Explicit time dependence adds $\langle\partial G/\partial t\rangle$ to the conservation equation, and an external symmetry-breaking potential generally permits exchange with the environment. Degenerate <eigenspaces> require choosing a compatible common basis rather than assuming every eigenvector already diagonalizes every commuting operator.
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