On the integrability cases of the equation of motion for a satellite in an axially symmetric gravitational field

The projection of an axially symmetric satellite's orbit on a plane perpendicular to the rotation axis (z=const.) is given by the second-order differential equation. $$frac{{y''}}{{1 + y'^2 }} = bar Psi _y - y'bar Psi _{x,}$$ where the prime denotes the derivative with respect tox and (bar Psi (x,y)) is a known function. Two integrability cases have been investigated and it has been shown that for these two cases the integration can be carried out either by quadratures or reduced to a first-order differential equation. Analytical and physical properties are expressed, and it is shown that the equation can be derived from the calssical plane eikonal equation of geometric optics.