  This paper shows that the Camassa-Holm (CH) spectral problem yields two
different integrable hierarchies of nonlinear evolution equations (NLEEs), one
is of negative order CH hierachy while the other one is of positive order CH
hierarchy. The two CH hierarchies possess the zero curvature representations
through solving a key matrix equation. We find that the well-known CH equation
is included in the negative order CH hierarchy while a Dym type equation is
included in the positive order CH hierarchy. Furthermore, under two constraint
conditions between the eigenfunctions and the potentials, the CH spectral
problem is cast in: (enumerate) a new Neumann-like N-dimensional system when it
is restricted into a symplectic submanifold of $\R^{2N}$ which is proven to be
integrable by using the Dirac-Poisson bracket and the r-matrix process; and a
new Bargmann-like N-dimensional system when it is considered in the whole
$\R^{2N}$ which is proven to be integrable by using the standard Poisson
bracket and the r-matrix process. (enumerate) In the paper, we present two
$4\times4$ instead of $N\times N$ r-matrix structures. One is for the
Neumann-like CH system (not the peaked CH system), while the other one is for
the Bargmann-like CH system. The whole CH hierarchy (both positive and negative
order) is shown to have the parametric solution which obey the constraint
relation. In particular, the CH equation constrained to some symplectic
submanifold, and the Dym type equation have the parametric solutions. Moreover,
we see that the kind of parametric solution of the CH equation is not gauge
equivalent to the peakons. Solving the parametric representation of solution on
the symplectic submanifold gives a class of new algebro-geometric solution of
the CH equation.
