Justice Mark Woolford set a hearing for this year to hear the applications. Rewa's trial date has been set for next February.
The application comes as Chambers raised concerns over fair trial rights because of the dramatised 90-minute television film about the Burdett case, In Dark Places, which will play on TVNZ 1 this year.
A stay of proceedings for a murder prosecution against Rewa had previously been applied by the Solicitor-General in 1998.
But last year the Deputy Solicitor-General, Brendan Horsley, on behalf of the Attorney-General, reversed the 1998 stay.
The stay was lifted after the Privy Council quashed Teina Pora's convictions for Burdett's rape and murder in 2015.
Pora was twice wrongly convicted for the crime and spent 22 years in prison. He has since received an apology from the Government and $3.5 million compensation.
Last month, Justice Geoffrey Venning, the Chief High Court Judge, also declined an application by Chambers for a judicial review of the decision to lift the stay.
"The reversal-of-stay document sets out the grounds for the decision, namely that given the quashing of Mr Pora's convictions no one has been held accountable for a murder, and the evidential sufficiency supporting the murder charge against Mr Rewa," Justice Venning said in his decision.
"There is no evidence before the court of any bad faith on the part of the Attorney-General in relation to the decision to lift the stay. The fact the effect of lifting the stay is that Mr Rewa will face a third trial is not, of itself, sufficient to provide justification for the court to review the Attorney-General's decision. Third trials, although rare, have been held in certain cases."
A stay has never been lifted in New Zealand's legal history.
Last month, Rewa was also denied parole - his first hearing before the New Zealand Parole Board.
The Parole Board's chairman, Justice Gendall, also directed that the written decision from the May 15 hearing will not be publicly released.
Rewa's next hearing has been set for April 2020, in line with the statutory cycle.