Is the Sun fixed in space?
By tracking the motion of sunspots across the Sun we can see it has rotational motion. It takes the Sun 31 days to rotate at its poles, but at its equator it moves faster, taking just 27 days to turn one revolution. This type of differential rotation is also present on the gas giants Jupiter and Saturn. The lack of a rigidly defined surface and a largely flowing composition means objects like the Sun can have varying rotational speeds as the whole body is not held together in the same way rocky planets like Earth are.
The Sun is also moving in two other ways. First, it is not stationary in the solar system; it is actually in orbit around every body that is also in orbit around it, such as all the planets. However, as the Sun is so massive its orbit is nominal. Indeed, the centre of mass of these orbital systems is often found within the Sun itself, so it only wobbles very slightly compared to the huge orbits of other celestial bodies. Beyond this, the Sun is also moving around the centre of the Milky Way along with the entire solar system; one complete orbit will take about 230 million years.
Answered by HIW.