Multi-component Painlevé ODE's and related non-autonomous KdV stationary hierarchies

First, starting from two hierarchies of autonomous Stäckel ODE’s, we reconstruct the hierarchy of KdV stationary systems. Next, we deform considered autonomous Stäckel systems to non-autonomous Painlevé hierarchies of ODE’s. Finally, we reconstruct the related non-autonomous KdV stationary hierarchies from respective Painlevé systems.