More details are expected to be announced on Tuesday, police said in Crete.
Eaton attended a conference at the Orthodox Academy when she apparently disappeared during a run.
Last Monday, her body was discovered by two locals According to Crete's police chief Konstantinos Lagoudakis, she was found deep in a cave.
It was located about 60 meters inside the cave under a duct covered by a large wooden pallet. The underground caves had been converted into a bunker by Nazi soldiers during World War II.
Police said that Eaton was suffocated. Minor puncture wounds were also found on her body, but the police said they were not considered a cause of death. The police believe the body was thrown into the cave because it was found hidden.
Lagoudakis told CNN on Thursday that he had never seen such a case in his four years as a police chief.
A really wonderful person. "
Eaton was the wife of British scientist Tony Hyman and mother of two sons.
Eaton's sister described her as a "consummate woman" with "insatiable curiosity" and praised her achievements.
"She loved the preparation of exquisite meals and an exotic sense of fashion, she loved perfume, she taught and practiced Tae Kwon Do as a second-degree black belt, ending crossword puzzles far too fast, playing concerts and playing well to Jane Austen's strictest description of a "perfected woman" while maintaining a natural modesty and an "insatiable curiosity"
Her sister added that Eaton often took care of it, She did not give enough time to her family and devoted herself to science.
She continued, "But anyone who read about their accomplishments in the field of Molecular and Developmental Biology or witnessed their joy in teaching their children, to comfort and inspire or share with and to love your husband, I would not have guessed. With a deep sensitivity and compassion she somehow made us all a priority. "
In a statement, the Max Planck Institute said Eaton was" an outstanding and inspiring scientist, a loving wife and mother, an athlete, and a truly wonderful person who has come a long way to our hearts.
The details of Eaton's death shocked the locals, some of whom thought for the first time that they might have died in a walking accident.
Ivana Kottosova and Kara Fox of CNN contributed to this report.  19659019]