Kepler's laws for planetary motion teaches us that the orbital equations for closed orbits must be elliptical in nature. More generally, orbits are conic sections, i.e., circular, elliptical, parabolical or hyperbolical.
Newtonian orbital equations are normally obtained by writing down Newton's equation of motion for an object moving in a gravitational field. The solution to the equation of motion gives the formula for a conic section.
The orbital formula is that of a circle for an eccentricity of zero, an ellipse if the eccentricity is less than unity, a parabola for eccentricity precisely unity and a hyperbola for eccentricity greater than unity.
Relativistic orbits are much more complex. The equations are usually obtained by writing down the formulas for a space-time geodesic and then partially solving the equations. There are no exact solutions for relativistic orbits, i.e., solutions enabling direct plots of the orbits do not exist.
There are various ways to obtain approximate solutions to the equations, depending on how relativistic the orbits become. Another popular way is to solve the equations numerically, i.e., step-by-step solutions, also called "numerical integration".
The pdf download will give you a very good feeling for the complexity of relativistic orbits and methods for solving them numerically. Also included is a quasi-Newtonian solution, using the acceleration equations from the chapter on the velocity effects on gravity.
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