Can a Hamiltonian commuting with a sector-label operator generate transitions between its sectors?

Suppose a Hilbert space has a direct-sum decomposition $\mathcal H=\bigoplus_{n\ge 0}\mathcal H_n,$ where the sectors are eigenspaces of a self-adjoint operator $\hat S_c$ , $\hat S_c|\psi_n\rangle=s_n|\psi_n\rangle.$ Let $\hat J$ be a self-adjoint generator of unitary evolution, $U(\tau)=e^{-i\hat J\tau},$ and suppose $[\hat J,\hat S_c]=0.$ A separate clock/phase operator $\hat S_t$ is introduced with the intended canonical relation $[\hat J,\hat S_t]=i\kappa.$ My question is: Can $\hat J$ , while commuting with $\hat S_c$ , generate genuine transitions $\mathcal H_n\longrightarrow\mathcal H_m,\qquad m\neq n,$ or must $U(\tau)$ preserve every eigenspace of $\hat S_c$ ? If the latter, what is the standard operator structure for introducing genuine sector-changing evolution while retaining a separate clock/phase operator? For example, would one require ladder/shift operators that do not commute with $\hat S_c$ ? I am particularly interested in the spectral/operator-algebraic issue rather than in a specific physical model.