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.

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 P_{IV} − P_{II} 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 P_{IV} hierarchy and the P_{II} hierarchy obtained here are different from those which have previously been given.

We consider a non-isospectral scattering problem having as its spatial part an energy-dependent Schrodinger operator. This gives rise to new completely integrable multicomponent systems of equations in (2 + 1) dimensions. Their reductions to systems in (1 + 1) dimensions have isospectral scattering problems and include multicomponent extensions of the AKNS equation and also a generalization of the Dym equation. An extension of the Fuchssteiner-Fokas-Camassa-Holm equation to (2 + 1) dimensions is also presented

The authors present a unified treatment of a modified singular manifold expansion method as an improved variant of the Painleve analysis for partial differential equations with two branches in the Painleve expansion. They illustrate the method by fully applying it to the Boussinesq classical system and the Mikhailov-Shabat system.

A complete Painleve test (1900) is applied to the generalized Burgers-Huxley equation using the version of the Painleve analysis recently developed by Weiss, Tabor and Carnevale (1983) for partial nonlinear differential equations. In so doing, the authors are able to find a complete set of new solutions as well as recovering some previous particular solutions already found by using ad hoc methods which have been recently published.