Although this question has been answered many times, I would like to add the following simple way to generate examples. It is clear that in the plane any central force, no matter how complicated, generates circular orbits as special cases. There are many ways to distribute matter in three space to generate such a force field. For example, any distribution of mass (which can, of course, consist of point masses) along the $z$-axis will produce such a field in the $x,y$-plane. As a side remark, such configurations occur in practice---e.g., as the gravitational field generated by a wire.

I might add for those interested in the theme of trajectories that there is a considerable body of work on the theme of which families of curves can occur as the trajectories of force fields, including a monograph "Differential-geometric aspects of dynamics", by Edward Kasner and his associates, much of which is easly accessible online. This is, however, not directly relevant to the theme of the query.

Although this question has been answered many times, I would like to add the following simple way to generate examples. It is clear that in the plane any central force, no matter how complicated, generates circular orbits as special cases. There are many ways to distribute matter in three space to generate such a force field. For example, any distribution of mass (which can, of course, consist of point masses) along the $z$-axis will produce such a field in the $x,y$-plane. As a side remark, such configurations occur in practice---e.g., as the gravitational field generated by a wire.

I might add for those interested in the theme of trajectories that there is a considerable body of work on the question of which families of curves can occur as the trajectories of force fields, including a monograph "Differential-geometric aspects of dynamics", by Edward Kasner and his associates, much of which is easly accessible online. This is, however, not directly relevant to the query.