Abstract
We give a new non-isospectral extension to 2 + 1 dimensions of the Boussinesq hierarchy. Such a non-isospectral extension of the third-order scattering problem 
xxx
+U
x
+(V
-
)
= 0 has not been considered previously. This extends our previous results on one-component hierarchies in 2 + 1 dimensions associated to third-order non-isospectral scattering problems. We characterize our entire (2 + 1)-dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non-isospectral scattering problem. This then allows an alternative approach to the construction of linear problems for the entire (2 + 1)-dimensional hierarchy. Reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems. In particular, we obtain a `fourth Painlevé hierarchy', i.e. a hierarchy of ordinary differential equations having the fourth Painlevéequation as its first member. We also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to Cosgrove.