Existence of symmetric waves in open metal--dielectric waveguides, a dielectric rod and the Goubau line, is proven by analyzing the of functional properties of the dispersion equations (DEs) and parameter-differentiation method, applied to the analytical and numerical solution of the DEs. Various limiting cases are investigated. Reduction to singular Sturm--Liouville boundary eigenvalue problemson the half-line is performed. Principal and higher-order surface waves  are investigated