The problem of diffraction of electromagnetic waves on a curvilinear surfaces with piecewise continuous admittance (impedance) boundary conditions, particularly on open-ended metallic surfaces is considered. It is shown that the coefficients of Fourier kind for some complete function system and the corresponding series for the function equal to the tangent component of the electric vector on the part of the surface where the admittance is finite and the surface electric current on the part where it is infinite, can be found by solving the inner and exterior boundary value problem. Each of the terms of the series satisfies the Meixner condition and the determined values of the tangent field on the given surface allow determining of the scattered field in the whole space. The method is illustrated with a sample of diffraction of a plane wave on a circular cylinder with the admittance periodically alternating in steps along the axis.