I assume that the 2 directors are the 2 x 50-50 shareholders.
Removal of the director can only be done in accordance with the Company Articles or in accordance with any service agreement. In the absence of any of that, directors can only be removed by a Special Resolution at an Extraordinary General Meeting of shareholders and the Director has to be given notice of the resolution and the opportunity to reply. If there are only two shareholders, each holding the same shares, then it’s virtually impossible to remove a director without a court application.
Regarding the change of trading name, (I assume you mean trading name and not the name of the company) there is no legislation with regard to a trading name and what it can and cannot be and whether it can be changed or not but the capacity to do that would be in any director service agreement. It is unlikely that the director has the power to do that.
To change the name of the company would require a special resolution and to be voted on by the shareholders.
Can I clarify anything for you?
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