A new analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of the KS elements, utilizing an analytical oblate exponential atmospheric density model. Due to the symmetry of the KS element equations, only one of the eight equations is integrated analytically to obtain the state vector at the end of each revolution. This is a uniqueness of the present theory. The series expansions include up to quadratic terms in e (eccentricity) and c (a small parameter dependent on the flattening of the atmosphere). Numerical studies are done with six test cases, selected to cover a wide range of eccentricity and semi-major axis, and a comparison of the three orbital parameters: semi-major axis, eccentricity and argument of perigee perturbed by the air drag with oblate atmosphere is made up to 100 revolutions with the numerically integrated values. The comparison is quite satisfactory. After 100 revolutions, with a ballistic coefficient of 50, a maximum difference of 39 metres is found in the semi-major axis comparison for a very small eccentricity (0.001) case having an initial perigee height of 391.425 km. One important advantage of the present theory is that it is singularity free, a problem faced by the analytical theories developed from the Lagrange's planetary equations. Another advantage is that the state vector is known after each revolution.