Yup, this is the type of strainer found on most Star snares, and the missing throw arm is a very common issue. The good news is that nothing is actually broken, you're just missing the parts. The small lever inside the strainer, near the top, with the hole in it is where a small threaded notch at the bottom of the arm fits in and is held in place by a square nut on the bottom side of the lever.
It's a simple design, but is obviously problematic because over time the nut can be worked loose with the vibrations of the drum being played. The nut falls to the floor of the garage/basement/bar, ditto the arm, both parts roll out of sight, and are never to be seen again. Now, without coming up with some new parts, you're left with an adjustable butt end. However, I'll bet a trip to the hardware store would provide you with some options to make this strainer usable again without having to drill for a new unit.
In this tread there's a pic of a complete example of this strainer:
http://www.vintagedrumforum.com/showthread.php?t=19566&highlight=astro