Abstract
We define new matrix fourth Painlevé hierarchies and a new matrix second Painlevé hierarchy. For our matrix fourth Painlevé hierarchies, properties such as auto-Bäcklund transformations, Bäcklund transformations and special integrals are derived, and known relations between fourth Painlevé transformations are extended to the matrix ODE hierarchies. The matrix ODE hierarchies presented here are derived as reductions of a new nonisospectral matrix dispersive water wave hierarchy and modifications thereof.
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