  Though completely integrable Camassa-Holm (CH) equation and
Degasperis-Procesi (DP) equation are cast in the same peakon family, they
possess the second- and third-order Lax operators, respectively. From the
viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic
and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great
difficulty in the construction of algebro-geometric solutions of the DP
equation. In this paper, we derive the DP hierarchy with the help of Lenard
recursion operators. Based on the characteristic polynomial of a Lax matrix for
the DP hierarchy, we introduce a third order algebraic curve
$\mathcal{K}_{r-2}$ with genus $r-2$, from which the associated Baker-Akhiezer
functions, meromorphic function and Dubrovin-type equations are established.
Furthermore, the theory of algebraic curve is applied to derive explicit
representations of the theta function for the Baker-Akhiezer functions and the
meromorphic function. In particular, the algebro-geometric solutions are
obtained for all equations in the whole DP hierarchy.

Exactly Solvable and Integrable Systems