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I think they will move in the same way we see them, that is, we will see fragments of an arc, but relative to the planet, also around us, but with a shifted center, but perhaps the movement will be more complex.

This movement cannot be described by the laws of physics in such a system. In the same system where the Earth revolves around the Sun, their movement is easily described – as a rotation around the Sun. This once again confirms the absoluteness of the Earth's movement around the Sun.

There is probably some mathematical equation for the movement of our Earth relative to the center of the Universe, and there is probably also an equation for the movement of the Sun. That's all. And to simplify calculations, scientists have come up with mathematical abstractions – the Sun rotating around the Earth, or vice versa, which is basically the same thing.

Theoretically, such an equation exists. For the Earth, such an equation consists of a number of factors. That is, the ABSOLUTE rotation around the Sun, the absolute rotation around the center of the galaxy, etc., are taken into account. As a result, we get an equation for the ACCELERATION VECTOR of the Earth at each moment in time (the equation of the trajectory does not say anything). For the Sun, it is the same, but only the factor of movement around the Earth is not included in its equation. Again, based on this, we can accurately establish that the Earth absolutely rotates around the Sun, and not vice versa, and this does not prevent the Earth-Sun system from also having accelerated movement.