We study the two-particle Schrödinger operators H (k)  with fixed quasimomentum of particles pair k T associated with the Fermi-Hubbard Hamiltonian H of a system of two identical fermions on the one-dimensional lattice Z with nearest-neighbor interaction  R . The existence or absence and the number of eigenvalues depending on the two particle quasimomentum and interaction energy is completely studied.