The linear homogeneous differential equation $z''=12yz$ with $y$ an arbitrary solution of Painlevé's first equation $y''=6y^2+x$ will be proved irreducible.
"Irreducibility of the Linear Differential Equation Attached to Painlevé's First Equation." Tokyo J. Math. 16 (1) 171 - 177, June 1993. https://doi.org/10.3836/tjm/1270128989