Physical and biochemical aspects of a proposed search for extraterrestrial eukaryotes (SETE) are considered. Such a program should approach the distinction between a primitive eukaryote and an archaebacteria. The emphasis on gene silencing suggests a possible assay suitable for a robotic investigation of eukaryoticity, so as to be able to decide whether the first steps towards eukaryogenesis have been taken in an extraterrestrial planet, or satellite. The experiment would consist of searching for cellular division and the systematic related delay in replication of heterochromatic chromosome segments. It should be noticed that the direct search for a membrane-bounded set of chromosomes does not necessarily determine eukaryotic identity, as there are prokaryotes that have membrane-bounded nucleoids. A closer look at the protein fraction of chromatin (mainly histones) does not help either, as there are some eukaryotes that may lack histones; there are also some bacteria as well as archaebacteria with histone-like proteins in their nucleoids. Comments on the recent suggestion of possible environments for a SETE program are discussed: the deep crust of Mars, and the Jovian satellite Europa, provided the existence of an ocean under its ice-covered surface is confirmed by the current Galileo mission.