American in Germany wants to move to Austria

Hi there, I live in Munich, Germany and run my own business (nature based holiday camps for children) and am a yoga teacher. I have a 3 year self-employed work visa for Germany. I want to move to Vienna, but will keep working in Germany. I would also like to work in Vienna as a yoga teacher and bring my business to Vienna as well. Is it possible to do this? Work in Munich and live in Vienna, legally? To work in Vienna, will I need permission or will my German arbeitserlaubnis suffice? If this is too complicated, where in Vienna can I go to sort this out, and perhaps receive permission to be self-employed? I meet all the legal requirements for Germany (taxes, health insurance, etc.). Thank you, Amie