On an extended second Painlevé hierarchy

We present a new extension of the second Painlevé hierarchy and study its properties. In addition to Lax pairs, Bäcklund transformations, auto-Bäcklund transformations and basic special integrals, we also consider a new phenomenon whereby we obtain relations between systems of different orders but of the same form. The extension made here of the second Painlevé hierarchy is based on the use of non-isospectral scattering problems and so is quite general. We thus expect to be able to obtain similar extensions of other Painlevé hierarchies, including not only for continuous examples but also for discrete and differential-delay examples. We believe that our work is also of relevance for Painlevé classification, since it gives information about classes of equation that may be of interest and in addition provides a key to the possible identification of equations isolated in such a process.