= Solution
With indices understood cyclically,
$$
\boxed{\dot P_i=\beta P_{i+1}+\alpha P_{i-1}-(\alpha+\beta)P_i.}
$$
The circulant generator has eigenvalues $0$ and $-3\phi\pm i\sqrt3\psi$, where $\phi=(\alpha+\beta)/2$ and $\psi=(\alpha-\beta)/2$. Expanding the initial vector $(1,0,0)$ in its three Fourier eigenvectors gives
$$
\boxed{P_1=\frac13[1+2e^{-3\phi t}\cos(\sqrt3\psi t)],}
$$
$$
\boxed{P_2=\frac13[1-2e^{-3\phi t}\cos(\sqrt3\psi t-\pi/3)],}
$$
$$
\boxed{P_3=\frac13[1-2e^{-3\phi t}\cos(\sqrt3\psi t+\pi/3)].}
$$
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