Statistical characteristics of light reflected by a rough random cylindrical homogeneous Gaussian surface are investigated using a modified method of specular points, as developed by Gardashov. In this proposed method, a special procedure for determining the light intensity near the caustics has been formulated. The probability distribution of the intensity of reflected light is expressed in terms of a special function, which is determined by the characteristic function of distribution of radii of curvature at the specular points and the distribution density of the number of specular points. The distribution of radii of curvature, derived by Gardashov, and expressed in terms of dimensionless radii of curvature, has a simple expression which does not contain any parameter of the surface (as a surface rms deviation, etc.). Consequently, it is universally valid and applicable to any cylindrical homogeneous Gaussian surface. After modification, the infinite dispersion of the reflected light intensity turns into a finite. The relationship between the distributions of reflected light intensity and the number of specular points, in the form of a Fredholm integral equation of the first kind is obtained. The kernel of the integral equation is expressed in terms of a characteristic function of the radii of curvature at specular points. The validity of formulae and relationships, thus derived, is tested by numerical simulations.