mersecske said:
Are there any satellite around the Moon on geostationary orbit?
The Earth is in a sense in a synchronous satellite of the Moon. We only see one face of the Moon, after all.
That's probably not what you are asking about, though. First let's look at lunar orbits that are direct analogs of geostationary orbits. Such orbits cannot not exist. A true geostationary (geo means Earth) orbit, has an orbital radius given by [tex]r^3 = GM_{\oplus}/{\Omega_{\oplus}}^2[/tex] where [tex]M_{\oplus}[/tex] is the mass of the Earth and [tex]\Omega_{\oplus}[/tex] is the Earth's rotational angular velocity. With some abuse of terminology, we can extend this meaning to any object, [tex]r^3 = GM/\Omega^2[/tex].
For the Moon this comes out to an orbital radius of 95,000 km. That is well beyond the Moon's sphere of influence (66,000 km) or its Hill sphere (61,000 km). Stable orbits only exist within 1/2 to 1/3 of the Hill sphere radius, so the highest stable orbit around the Moon is somewhere in the 20,000 to 30,000 km range. So a direct analog of an Earth synchronous orbit is not possible for the Moon.
So, let's broaden the meaning of a synchronous orbit to mean any orbit about some body such that the same side of that body is all that is seen. Because the Moon is tidally locked to the Earth, the Earth is in a sense a synchronous satellite of the Moon. That might seem a bit snide, but its not. This expanded definition means that a satellite in one of the Earth-Moon Lagrange points can be viewed as being in a synchronous orbit about the Moon.
There are none yet. However, the concept of an Earth-Moon L1 space station is quite appealing. The idea keeps coming up as a possibility in NASA and elsewhere.