Trying to model cartoon eyes, need help

I’m trying to model an MLP style pony and have been rather successful with the body. However, I have no idea how to handle the eyes.

They should move like this: This appears to be Autodesk Maya, however, and I have no idea how they did this.

My progress I modeled some basis eyes, but these are not suitable for animating. a few poses

I do already know how to track spherical eyes, but this complete stumped me. Is there some tutorial somplace, or can someone make one? It looks as if the eye is a single sphere with an adjustable texture of some sort.

Hope this can help you, good luck.