Planar motion is the result of vectors summing to zero. In the case of force vectors it is the equilibrium condition: 0 = f1 + f2 + f3 + ..fn. Whenever the sum of the vectors is zero, the motion is confined to a plane thus planar motion. When there are three vectors the sum forms a triangle. If there are two vectors the sum forms a line in the null case the sum forms a point at the origin summing to zero
Conservation of energy is a zero force condition, thus giving planar motion:
Derivative of Energy= force = 0 at Conservation of Energy, therefore conservation of Energy motion is planar motion. Gravitational orbits are planar motion orbits due to conservation of gravitational energy.