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For those playing at home, and in NSW, you want to fill in this form and take it to a Service NSW office with either your Company Certificate of Registration, or an except which you can download from your ASIC login...
I believe that a "Business Authority to Register" is literally a letter signed by someone with authority (like a director) to allow you to act on behalf of the company. The person signing the letter could be you if you are a director. I have had to do this from time to time with other companies...