In this article, the orbit of a satellite placed in a rotating gravitational (Kerr new metric) field is derived by using symmetries of the metric method. It reveals that the orbit is an ellipse depending on mass, angular momentum of the Earth and the initial inclination angle. A mathematical treatment analyzes the orbit of a satellite around the Earth at an inclination of the equatorial plane. The advance of perigee of the orbital motion is administered. Additionally, the periodic time and the advance of perigee for some artificial satellites are calculated. It is shown that the advance of perigee of the orbital motion of the satellite is inversely proportional to its periodic time. The perturbed differential equation, describing the orbit of the satellite, is utilized to govern stability criterion. Furthermore, the effects of mass, angular momentum of the Earth, and the semi-latus rectum of the elliptical orbit on the satellite motion are discussed. As a result of examining perturbed geodesics, the orbit stability problem is inspected. A new theoretical finding related to the stable satellite model, consistent with those of the previous works, is obtained. Moreover, a formulation of the precession of the nodes has been accomplished.