We show that certain partial differential equations associated to nonisospectral scattering problems in $\text{2+1}$ dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations. This is illustrated using (an extension of) a known second-order and two new third-order nonisospectral scattering problems. These scattering problems allow us to derive new hierarchies of integrable partial differential equations, in both $\text{1+1}$ and $\text{2+1}$ dimensions, together with their underlying linear problems (isospectral and nonisospectral); and also new hierarchies of integrable ordinary differential equations, again with their underlying linear problems.

The second Painleve hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well known second Painleve equation, P-11. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painleve analysis to ordinary differential equations. We extend these techniques in order to derive auto-Backlund transformations for the second Painleve hierarchy. We also derive a number of other Backlund transformations, including a Backlund transformation onto a hierarchy of P-34 equations, and a little known Backlund transformation for P-11 itself. We then use our results on Backlund transformations to obtain, for each member of the P-11 hierarchy, a sequence of special integrals