We study a family of real rational functions with prescribed critical points and the evolution of its poles and critical points under particular Loewner flows. The recent work by Peltola and Wang shows that the real locus of these rational functions contains the multiple SLE$(0)$ curves, the deterministic $kappa to 0$ limit of the multiple SLE$(kappa)$ system. Our main results highlight the importance of the poles of the rational function in determining properties of the SLE$(0)$ curves. We show that solutions to the classical limit of the the null vector equations, which are used in Loewner evolution of the multiple SLE$(0)$ curves, have simple expressions in terms of the critical points and the poles of the rational function. We also show that the evolution of the poles and critical points under the Loewner flow is a particular Calogero-Moser integrable system. A key step in our analysis is a new integral of motion for the deterministic Loewner flow.
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