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The planets in the Solar System (and, for the most part, their moons as well) actually are arranged in more or less a 2-dimensional plane that way. That is NOT because of how the Sun distorts the spacetime around it, but rather because of the way gas clouds tend to form into star systems. The gas clouds' own gravity tends to pull them from a spherical shape into a disc shape before the planets form, so when the planets do form they end up close to a plane. That said, there are other smaller objects orbiting the Sun, such as comets and KBOs, which have orbits way outside that plane, because they formed differently from the planets.
