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