Rhys and Shaun Scott, who were two years old, were found in the garden tank in Dalgety Bay on Saturday at about 08:20.
They were later pronounced dead at hospital in Kirkcaldy.
The investigation is now under the direction of the Scottish Fatalities Investigation Unit (SFIU).
The Crown Office said the family would be kept informed of any developments.
A spokesperson said: "These tragic deaths are the subject of investigation and a report to the procurator fiscal."
It is believed that the family rented the home from a man who formerly ran a business from there, selling ornamental koi carp.
In 2012, Fife Council gave planning permission for the "installation of a fish tank and enclosure to the side" of the house, at an estimated cost of Â£25,000.
A police report into the deaths of twin brothers who drowned in a large fish tank at their family home in Fife has been handed to the procurator fiscal.