We find for the first time that a model Hamiltonian of s-wave superconductors in the presence of spin-orbit interactions and a Zeeman field is exactly solvable.Most intriguingly,based on the exact solutions,a novel type of Fulde-Ferrel-Larkin-Ovchinnikov ground state is rigorously revealed,in which the center-of-mass momentum of the fermion pair is proportional to the Zeeman field.We also generalize our exact analysis to the spin-orbit-coupled Bose-Einstein condensate.