Hi there and
to you both!
I am on a variation of Atkins, the book
The Idiot-proof Diet by Neris and India. I've changed that up even more by adding a heavier exercise plan.
I've been on my journey since May 14th. I exercise 6 days a week and eat low carb. Yes, it is possible to do both (just drink LOTS of water...and the funny thing is, the more you drink, the more you want!).
So, to answer your question, yes, you can run while on Atkins. It's what I do the most of also (was a track runner in jr high/high school...one would think I would know better than to get this big).