Neumann Type Integrable Reduction to the Negative-Order Coupled Harry--Dym Hierarchy

Based on the Lax compatibility, the negative-order coupled Harry--Dym (ncHD) hierarchy depending upon one parameter $\alpha$ is retrieved in the Lenard scheme, which includes the two-component Camassa--Holm (2CH) equation as a special member with $\alpha=-\frac14$. By using a symmetric constraint, it is found that only in the case of $\alpha>1$ the ncHD hierarchy can be reduced to a family of backward Neumann type systems by separating the temporal and spatial variables on the tangent bundle of a unit sphere. The resultant backward Neumann type systems are proved to be completely integrable in the Liouville sense via a Lax equation. Finally, for $\alpha>1$, the relation between the ncHD hierarchy and the backward Neumann type systems is established, where the involutive solutions of backward Neumann type systems yield the finite parametric solutions to the ncHD hierarchy.