Isomonodromy and Painlevé Type Equations, Case Studies

There is an abundance of equations of Painlevé type besides the classical Painlevé equations. Classifications have been computed by the Japanese school. Here we consider Painlevé type equations induced by isomonodromic families of linear ODE's having at most ${z=0}$ and $z=\infty$ as singularities. Requiring that the formal data at the singularities produce isomonodromic families parametrized by a single variable $t$ leads to a small list of hierarchies of cases. The study of these cases involves Stokes matrices and moduli for linear ODE's on the projective line. Case studies reveal interesting families of linear ODE's and Painlevé type equations. However, rather often the complexity (especially of the Lax pair) is too high for either the computations or for the output. Apart from classical Painlevé equations one rediscovers work of Harnad, Noumi and Yamada. A hierarchy, probably new, related to the classical $P_3(D_8)$, is discovered. Finally, an amusing “companion” of $P_1$ is presented.