A nonlinear integral equation is obtained for the static structure factor of an electron liquid. The equation is exact both in the small- and large-wave-vector limits. It has been solved self-consistently and values of the structure factor thus obtained are used to calculate the pair-correlation function of the system. The results are found to be almost positive over the entire metallic density range. An integral equation for the magnetic structure factor is also obtained which is exact for large wave vectors. The spin pair-correlation functions of the electron liquid are calculated using the self-consistent values of the magnetic structure factor obtained by solving the integral equation. The quality of various spin pair-correlation functions is quite good as compared to those obtained from earlier calculations.