A method for obtaining an explicit closed-form uniform asymptotic approximation of the wave function for the scattering by a rotationally symmetric potential is presented. This involves a systematic procedure to generate two orthogonal variables (u,v), where u is the similitude variable for the eikonal equation and is used as variable for the etalon function. The difference of the two orthogonal variables must be z in order to have a plane wave solution in the absence of the potential. The new Ansatz based on an etalon function generated by the problem itself is used. It allows removing the singularities at caustic surfaces of the 3D WKB wave function approximation. The method is illustrated in the case of a certain non-central rotationally symmetric potential. The uniform asymptotic approximation of the wave function, as well as the scattering amplitude containing the poles are obtained.