But I think you with a so called address change, which you can find on eg SKVs homepage, will get a paper sent to your new address confirming your change of residency which then should regulate your specific situation. I am sure I could just have moved all other liquid means over to the US too, no questions asked. This will then cover you for 1 year as they do not issue it for longer periods. You should make sure to get a local social security number. But your CY TIN or what they call there, should be fast but they will want a local adress, temporary or not, which will probably be fast.
Alternatively, take a 2 month vacation to eg CY starting June 29 this year till September.
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Can they maintain an apartment Daniel - and note they have only lived in SE for 5 years? Another alternative is to move in September, if this month is magical, but do not pay yourself any salaries - take it out instead when you would have permanent residency in CY anyways. The June to September Vacation is a good idea. But now you got more advice than you asked for. Alternatively, just wait for SKV to send you your tax form and accept the little you will have to pay for interest H! There are even horror cases about this.