This may not be the answer that you are looking for, but I believe that you should be able to write the Painlevé VI equation as a hamiltonian system, in which case it would govern the dynamics of some "physical" system.  The reason for the double quotes is that this is perhaps not a system arises in nature.  Most likely -- although I don't know for sure -- it will not be just coupled harmonic oscillators.

Less directly, Painlevé equations arise in the study of integrable hierarchies, some of which (e.g., KdV, nonlinear Schrödinger,...) are used to model natural phenomena.