Is there any known planets or star systems in interstellar outer space that exist though not in motion?
All objects are subject to Newton's law of gravitation, meaning that anything, even billions of light years away, will tug on an object. The universe is not isotropic on small scales, which is important meaning that the object cannot be stationary. Also, thermodynamics states that all particles move when not at absolute zero, which cannot be attained no matter what.
Expanding Zephyr's comment to an answer: what is in motion and not in motion is not universally agreed upon. If you are sitting on a chair beside a road, the cars will be in motion while you are not. However, if you are driving a car looking at someone on a chair beside the road, then you are stationary and they are in motion. In physics jargon we call the two viewpoints as different reference frames, and the Principle of Relativity says that the laws of physics hold in all reference frames, i.e. both reference frames are equally valid.
Accordingly the question "are there any planets or star systems that are not in motion" is not really answerable. You can always define a reference frame such that the planet or star system is not in motion, just like you can always define a reference frame such that they are in motion.
It's a good question!
In free space without any friction or anything else to "hang on to" the only thing that keeps everything from falling into everything else is called "conservation of energy".
Once the gravity between objects starts pulling them together, they loose potential energy so they pick up kinetic energy.
That plus conservation of angular momentum means they end up orbiting each other most of the time. Most things we see in space are orbiting something else, or perhaps several other things.
Since the distance scales are so huge compared to what we are familiar with on Earth or even in our own solar system, the motion is very very slow on our timescale.
But if we could speed things up and see what happens in a million or billion years, the sky would look pretty crazy and the stars would be moving all over the place.
We live in a snapshot. Our lives are shorter than a flash.
- this answer to Are there places in the Universe without gravity?
- Density and dimensionality of zeros in inverse square force fields of randomly distributed sources in (at least) 1, 2 and 3 dimensions?
there is an abstract discussion of mathematical points where there really could be no motion for a short time.
But since everything is moving, those points move too, so there won't be static solutions.
If we throw a ball up in the air, for an instant it seems motionless at the top of the arc. But that moment lasts for zero time. It looks motionless for longer than zero only because we can't easily detect the motion.