Considered the law of the gravitation, the bodies in translation would not have to describe a perfect orbit?

Answer 1

Depends what you describe as a perfect orbit.

Orbits are rarely perfect.

Earth's orbit is not circular, it is elliptical. In July, we are furthest from the Sun and in January, we are closest.

Because of perturbation from other planets, we do not retrace the same orbit year after year (our ellipse is not closed). Presently, our average orbital distance is increasing by 400 km per year and it will soon be 1 A.U. (over the last decades, we were a tiny bit closer than our long-term average).

The Sun is losing mass (because it is emitting energy -- E = mc^2 and all that). Our extremely-long-term orbital distance is creeping outwards (in centimetres per century).

Of course, it is the barycentre of the Earth-Moon system that is in orbit around the Sun. The Earth's path really wobbles a little more than 12 times a year (by thousands of km) relative to the path of the barycentre.

If, from the comfort of a spacecraft sharing an orbit with the Earth-Moon system (let's say at 1 A.U.), we were to 'toss' an object and give it some translation velocity, it would simply take up a new orbit around the sun. This new orbit would include at least one point that is at 1 A.U. It might be more elliptical or less elliptical than our orbit (depending on the speed differential that we would have given to the new artificial planetoid)

Answer 2

It might be a many bodies problem in a gravitational field.