More details are expected to be announced on Tuesday, the Criminal Police reported.
Eaton attended a conference at the Orthodox Academy when she disappeared, apparently during an escape.
Last Monday, her body was discovered by two local residents. Deep in the cave, according to the Chief of the police of Crete Konstantinos Lagoudakis
It was found about 60 meters (about 200 feet) inside the cave under an air shaft covered with a large wooden pallet. Underground caves were turned into a bunker by Nazi soldiers during the Second World War.
The police say Ethan was suffocated. Minor wounds have been discovered on her body, but police say she is not believed to be the cause of her death. Police believe the body was dumped in the cave because it was exposed face down.
Lagoudakis told CNN on Thursday he never saw such a case in his four years as a police chief.  Really wonderful man "
Eaton is the wife of British scientist Tony Heyman and mother of two sons, according to the institute
Eaton's sister described her as "a complete woman" of "insatiable curiosity" and praised her achievements. "She has a great pleasure in preparing fine dishes and has an exotic sense of fashion, she loves perfume, and she teaches and practices Tae Kwon Do as a second-degree black belt, too quickly crossing crossword puzzles, playing concerts, and responding to the strictest a description of Jane Austin as a "complete woman" while maintaining natural humility and "insatiable curiosity," her sister wrote.
Her sister added that Ethan would often worry that he would not give his family enough time
She continued: "But everyone who reads for no or who has witnessed her joy in lessons, comforting and inspiring her children, or sharing with, and with deep sensitivity and compassion, she somehow made us a priority. "
In a statement, The Max Planck Institute said Ethan was "an outstanding and inspiring scientist, a loving husband and mother, an athlete, and a truly wonderful man loved by all of us."
The details of Ethan's death shook the locals, some of whom they thought she could die at Traveler with hiking.
Ivana Kotosova and Kara Fox from CNN contributed to this report.