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Abstract.The second Painlevé 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 Painlevé equation, P II .In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive autoBäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformation onto a hierarchy of P 34 equations, and a little known Bäcklund transformation for P II itself.We then use our results on Bäcklund transformations to obtain, for each member of the P II hierarchy, a sequence of special integrals.

We present a generalized non-isospectral dispersive water wave hierarchy in 2 + 1 dimensions. We characterize our entire hierarchy and its underlying linear problem using a single equation together with its corresponding non-isospectral scattering problem. This then allows a straightforward construction of linear problems for the entire generalized 2 + 1 hierarchy. Reductions of this hierarchy then yield new integrable hierarchies in 1 + 1 dimensions, and also new integrable hierarchies of ordinary differential equations, all together with their underlying linear problems. In particular, we obtain a generalized PIV − PII hierarchy; this includes as special cases both a hierarchy of ODEs having the fourth Painlevé equation as first member, and also a hierarchy of ODEs having the second Painlevé equation as first member. All of these hierarchies of ordinary differential equations, as well as their underlying linear problems, are new; both the PIV hierarchy and the PII hierarchy obtained here are different from those which have previously been given.

We show that certain partial differential equations associated to nonisospectral scattering problems in 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 1+1 and 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.

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