2 Added a reference for the hamiltonian for Painlevé VI

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.

Edit

An explicit form for the Hamiltonian of Painlevé VI can be found here, right after Theorem 2.1. Although it is polynomial, it does depend explicit on 'time'. Hence as a hamiltonian system it is certainly not very natural. For one thing, energy is not conserved. This is to be expected, since Painlevé VI is itself not integrable, which it would have to be if you could find a conserved quantity as it is a one-dimensional system.

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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.