= Solution
The orbiting planet produces a potential $V(r,\vartheta,\varphi-\Omega t)$. It has explicit time dependence and exerts an azimuthal <torque>, so generally $\dot E=V_t$ and $\dot L_z=-V_\varphi$ are nonzero. Nevertheless a simultaneous <time-translation symmetry> $\delta t=\lambda$ and <rotation> $\delta\varphi=\Omega\lambda$ leaves the <Lagrangian> and $dt$ unchanged. <Noether's theorem> gives $Q=\Omega p_\varphi-E$ and hence
$$
\boxed{E_J=E-\Omega L_z=\text{constant}.}
$$
Equivalently $V_t=-\Omega V_\varphi$ implies $d(E-\Omega L_z)/dt=0$. This <Jacobi integral> assumes a rigidly rotating potential with constant $\Omega$ and negligible back-reaction from the <test particle>.
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