1. I regret to say tha t you need to set up a new corporate structure with a different name for the venture with the new IP. There is no legal way you are going to get rid of 2 Director/shareholders with 50% of the share capital of a company. There is an unresolvable impasse. So, you need to move quickly on and set up a new company with a different name to handle the new IP. You should put the new IP into this new venture and avoid all contamination with the former venture with the old App.
2. The reason for the new name is ***** ***** need to avoid any action for the civil wrong of passing off which allows a company to sue for someone using the goodwill of its business to generate business for itself. You should also both resign as directors of the existing company before you set up the new company. This is because directors owe fiduciary duties to a company and its shareholders, so if they act or set up a rival company they can have this taken off them if they compete or use the opportunity of the company the are a director of, to set up in business.
3. So, clear water must be established between the old company with the Old App and the new company with the new IP. So new name, resign directorships, and a new company must be created. Ultimately, you can come back and purchase the old app if you wish. But wait until your new business is established.
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Answer 1. The liability to an action for passing off arises whenever there is a risk of confusion amongst customers as to the business selling a product such that one company is trading on the other company's goodwill. So, by adopting the same name, you run the risk that customers of the new US company will confuse it with the old UK company. There is no problem is you have no customers of the old company in the UK. However, there have been cases involving the name Budweiser in Europe where any sales giving rise to goodwill can lead to an action for passing off.
Erratum - The Answer 1 should read "There is no problem is you have no customers of the old company in the US"
Answer 2. My suggestion of buying the Old App is contingent upon the two bad directors ultimately wanting to sell. Essentially six months after the two good directors exit, the old App company will lose its fizz as there will be no one running its affairs properly. At this point a sale will become attractive. However, it requires the co-operation of the two bad directors.
Answer 2 (continued) It is not suggested that the IP belonging to the old App can be "lifted" out of the old App company. In stating that a sale can take place, it is an iron rule of business that the most competitive survive. So, if as you suggest, the bad directors are not pulling their weight, a sale will be attractive.