Slow Motor that does half rotation, stops for 20 Seconds, returns.

What kind of motor are you talking about?  If it's a common can motor, replace it.  If it's a Pulmore AC motor, I'd start with cleaning up the commutator and giving it new brushes.

GRJ, Will your track signal driver make a motor do what he's askin?  Moed321, what are you trying to accomplish?  What's the application?  Although I guess it doesn't matter if the device you're asking for does what the title sez. 

I totally misread it, I thought he had a malfunctioning motor.

My solution would be a model airplane servo.

Hard to believe something like that would be available off-the-shelf. 

I'm thinking you need to select a geared DC-motor and attach 2 angle sensors (limit switch, Hall switch, etc.) to detect the start/end position and the half-rotation position.  Then cobble together a circuit using timer/relay modules to sequence the action...or use an Arduino or similar if comfortable working with programmable controllers.

As asked above, can you share more about what you're trying to accomplish?  If the specifications are somewhat loose such as half rotation means about half rotation, 20 seconds means about 20 seconds, etc., then other options come into play such as perhaps using a servo-motor which has integral position sensing.

You might also looking into adapting an off-the-shelf trolley control module with 2-station out-and-back operation.  In other words it applies power to the track, trolley goes from station 1 to station 2, stops, waits some settable time, reverses direction and returns to station 1.

I thought he had something wrong too John.....I was like ....what in the world?....god I'm getting old...

"Be specific and you will be terrific."  Dale Carnegie