As a NZ citizen, your partner has an automatic right to live and work here. As a NZ PR, you don't have that automatic right, but as the partner of a NZ citizen who is in Australia on a Special Category Visa (SCV), you can apply for a New Zealand Citizen Family Relationship (Temporary) Visa (Subclass 461) which is a temporary visa that allows you to live and work in Australia. It is a 5 year temporary visa but is renewable if you meet certain residence and family requirements even if you are no longer with your partner.
Your New Zealand citizen partner automatically gets a Special Class Visa (SCV) on entry into Australia (after showing their passport). The SCV allows them to live and work in any occupation in Australia for as long as they want. You can apply for a partner-offshore visa which will allow you to live and work in Australia too,
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As a NZ citizen, your partner has an automatic right to live and work here. As a NZ PR, you don't have that automatic right, but as the partner of a NZ citizen who is in Australia on a Special Category Visa (SCV), you can apply for a New Zealand Citizen Family Relationship (Temporary) Visa (Subclass 461) which is a temporary visa that allows you to live and work in Australia. It is a 5 year temporary visa but is renewable if you meet certain residence and family requirements even if you are no longer with your partner.
http://www.immi.gov.au/migrants/family/461/
Yes, you both can.
Your New Zealand citizen partner automatically gets a Special Class Visa (SCV) on entry into Australia (after showing their passport). The SCV allows them to live and work in any occupation in Australia for as long as they want. You can apply for a partner-offshore visa which will allow you to live and work in Australia too,