This work studied the local well-posedness of the Cauchy problem for a one-dimensional variational wave system arising from the theory of nematic liquid crystals. The system is degenerate and singular at spatial infinity. We applied the method of characteristics and the contraction mapping principle to establish the local existence and uniqueness of classical solutions. A suitable weighted $L^{\mathrm{\infty }}$ norm was introduced to ensure that the solution has no loss of regularity and the existence time of the solution is independent of the spatial variable.