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The geometry of the real planar polynomial differential systems having their orbits embedded in conics

We classify and provide the global phase portraits in the Poincaré disc of all real planar polynomial differential systems having their orbits embedded in conics. This is achieved via the real affine classification of the pencils of conics, and each type corresponds to an equisingularity type of pencil. All such polynomial vector fields have degree less than or equal to 3. Also, when the degree is 3, infinity is filled with singular points.