= Solution
In a <quantum-trajectory unravelling>, a pure state evolves between jumps under
$$
H_{\rm eff}=H-\frac i2\sum_\alpha L_\alpha^\dagger L_\alpha,
$$
and a jump of type $\alpha$ sends $|\phi\rangle$ to $L_\alpha|\phi\rangle/\|L_\alpha|\phi\rangle\|$. Averaging $|\phi_t\rangle\langle\phi_t|$ over the stochastic jump records recovers $\rho(t)$.
Equivalently, over a short interval $dt$ use the operators in a <Kraus representation>
$$
K_0=I-iH_{\rm eff}dt,
\qquad
K_\alpha=\sqrt{dt}\,L_\alpha,
$$
and the <Stinespring dilation> $V=\sum_\mu K_\mu\otimes|\mu\rangle_E$. Iterating with fresh environment systems produces a pure system-environment history whose <partial trace> is the Lindblad evolution.
If the fixed point is pure, stationarity requires $L_\alpha|\psi_*\rangle=\ell_\alpha|\psi_*\rangle$ for every $\alpha$, together with preservation of its ray by the adjusted effective Hamiltonian. After shifting the jumps one may take it to be a <dark state of a Lindblad equation>. Once a trajectory reaches that ray, neither no-jump evolution nor a jump takes it away; in the example of part (c), a jump transfers any excited component directly into the singlet.
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