  We provide a construction of the two-component Camassa-Holm (CH-2) hierarchy
employing a new zero-curvature formalism and identify and describe in detail
the isospectral set associated to all real-valued, smooth, and bounded
algebro-geometric solutions of the $n$th equation of the stationary CH-2
hierarchy as the real $n$-dimensional torus $\mathbb{T}^n$. We employ
Dubrovin-type equations for auxiliary divisors and certain aspects of direct
and inverse spectral theory for self-adjoint singular Hamiltonian systems. In
particular, we employ Weyl-Titchmarsh theory for singular (canonical)
Hamiltonian systems.
  While we focus primarily on the case of stationary algebro-geometric CH-2
solutions, we note that the time-dependent case subordinates to the stationary
one with respect to isospectral torus questions.
