Being one of the important parts of fundamental mathematics, Model Theory is a young subject for modern researchers in this area. However, according to the last obtained results, this discipline will play a crucial role in the future of mathematical science. As well-known, the name "Model Theory" was introduced in 1954. It is important to distinguish that classical Model Theory introduces concepts based on considering complete theories. The given article is dedicated to the research of Jonsson theories of unars. Jonsson theories are, generally speaking, not complete. Hence, the results obtained in this article are strengthened. Firstly, we considered the theory of all unars and a class of existentially closed models of this theory. Secondly, we expanded the signature of unars that contains only one unary functional symbol by a new unary predicate and constants. Thirdly, we obtained some results concerning the universals and primitives of considered theory’s existentially closed Jonsson unars. Since we are using the new methodology (so-called semantic method) for the research of Jonsson existentially closed unars. Semantic methods consist of transferring properties of fixed complete theory to considered Jonsson theory.