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We describe the Loewner chains of the real locus of a class of real rationalfunctions whose critical points are on the real line. Our main result is thatthe poles of the rational function lead to explicit formulas for the dynamicalsystem that governs the driving functions. Our formulas give a simple methodfor mapping the class of rational functions into solutions to a non-trivialsystem of quadratic equations, and for directly showing that the curves in thereal locus satisfy geometric commutation and have the geodesic multichordproperty. These results are entirely self-contained and have no reliance onprobabilistic objects, but make use of an integral of motion for the Loewnerchain that is motivated by ideas from conformal field theory. We also show thatthe dynamics of the driving functions are a special case of the Calogero-Moserintegrable system, restricted to a particular submanifold of phase space carvedout by the Lax matrix. Our approach complements a recent result of Peltola andWang, who showed that the real locus is the deterministic kappa to 0 limit ofthe multiple SLE(kappa) curves.

Pole dynamics and an integral of motion for multiple SLE(0)