  Recently, scattering of a Klein-Gordon particle in the presence of mixed
scalar-vector generalized symmetric Woods-Saxon potential was investigated for
the spin symmetric and the pseudo-spin symmetric limits in one spatial
dimension. In this manuscript, the bound state solutions of the Klein-Gordon
equation with mixed scalar-vector generalized symmetric Woods-Saxon potential
are examined analytically within the framework of spin and pseudo-spin symmetry
limits. We prove that the occurrence of bound state energy spectrum exists only
in the spin symmetric limit, while in the pseudo-spin symmetric limit, the
bound state spectrum does not exist. Besides the theoretical proof, the
Newton-Raphson numerical methods are used to calculate the bound state energy
spectra of a neutral Kaon particle, confined in a generalized symmetric
Woods-Saxon potential, energy well constituted with repulsive or attractive
surface interactions, for the spin and pseudo-spin symmetric limits,
respectively. Numerical results are consistent with the non-existence of the
bound state energy spectrum in the pseudo-spin symmetric limit.
