  We investigate a one-dimensional nonlinear wave system which arises from a
variational principle modeling a type of cholesteric liquid crystals. The
problem treated here is the Cauchy problem for the same wave speed case with
initial data on the parabolic degenerating line. By introducing a partial
hodograph transformation, we establish the local existence of smooth solutions
in a weighted metric space based on the iteration method. A classical solution
of the primary problem is constructed by converting the solution in the partial
hodograph variables to that in the original variables.
