  In this paper, we present a systematical account of the descending procedure
from six-vertex model to the $N$-state chiral Potts model through fusion
relations of $\tau^{(j)}$-operators, following the works of Bazhanov-Stroganov
and Baxter-Bazhanov-Perk. A careful analysis of the descending process leads to
appearance of the high genus curve as rapidities' constraint for the chiral
Potts models. Full symmetries of the rapidity curve are identified, so is its
symmetry group structure. By normalized transfer matrices of the chiral Potts
model, the $\tau^{(2)}T$ relation can be reduced to functional equations over a
hyperelliptic curves associated to rapidities, by which the degeneracy of
$\tau^{(2)}$-eigenvalues is revealed in the case of superintegrable chiral
Potts model.
