Nurowski's conformal class of a maximally symmetric (2,3,5)-distribution and its Ricci-flat representatives

We show that the solutions to the second-order differential equation associated to the generalised Chazy equation with parameters $k=2$ and $k=3$ naturally show up in the conformal rescaling that takes a representative metric in Nurowski's conformal class associated to a maximally symmetric $(2,3,5)$-distribution (described locally by a certain function $\varphi(x,q)=\frac{q^2}{H''(x)}$) to a Ricci-flat one.