My wife and son are Lithuanian citizens and I am a New Zealand citizen with a Lithuanian residence permit.
Can I work alone in another EU state without my partner joining me for a few months, or will I need to apply for a work visa for that state?
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In general, the free movement directive provides that you have freedom of movement when you accompany or join your EU wife. You therefore cannot rely on it to secure work authorization in a country where she is not.
Furthermore, because you cannot benefit from free movement without your wife, your Lithuanian residence permit authorizes you to spend no more than 90 days in any 180-day period in Schengen countries other than Lithuania without her, so you also need to worry about residence authorization.
Depending on the nature of your work, it may implicate the free movement of services, but I do not know what consequences this has for work or residence authorization.
Your safest course of action will therefore be to seek work authorization in the EU state where you hope to work, and, if you will be there for more than 90 days, a long-term visa. Without knowing which state it is, it is impossible to give a more specific answer.