The application of the generator coordinate method is extended to non-harmonic systems. The many-dimensional Hill-Wheeler integral equation is reduced to a one-dimensional integral equation by expressing all independent parameters in the generator function by a single parameter. It is shown that the subspace spanned by a proper single-parameter family is the same as that spanned by the many-parametric family of generator functions. The subspace of the seniority-zero states relevant to the description of pairing vibrational states is spanned by the N-projected BCS function depending on the gap parameter. It is shown on a schematic three-level system that a good description of energies and two-body transfer amplitudes of the pairing vibrational states can be obtained in a subspace much smaller than the seniority-zero subspace. Having a method working equally well for superconducting nuclei, non-superconducting nuclei and in the intermediate region, we tested how well the seniority-zero subspace is suitable for the description of low-lying 0 + states in Pb and Sn isotopes. It seems to us that only a qualitative description is possible.