On a new non-isospectral variant of the Boussinesq hierarchy

We give a new non-isospectral extension to 2 + 1 dimensions of the Boussine sq hierarchy. Such a non-isospectral extension of the third-order scatterin g problem psi(xxx) + U psi(x) + (V - lambda)psi = 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 p roblems. We characterize our entire (2 + I)-dimensional hierarchy and its l inear problem using a single partial differential equation and its correspo nding non-isospectral scattering problem. This then allows an alternative a pproach to the construction of linear problems for the entire (2 + 1)-dimen sional hierarchy. Reductions of this hierarchy yield new integrable hierarc hies of systems of ordinary differential equations together with their unde rlying linear problems; In particular, we obtain a 'fourth Painleve hierarc hy', i.e. a hierarchy of ordinary differential equations having the fourth Painleve equation as its first member. We also obtain a hierarchy having as its first member a generalization of an equation defining a new transcende nt due to Cosgrove.