The Korean government has declared a transition from e-government to AI-based intelligent government through policy master plans, such as the 2nd E-Government Basic Plan (2021-2025) and the National AI Everyday Life Action Plan (2023). This is largely due to the rapid spread of generative AI, represented by ChatGPT, in everyday life, as a significant number of citizens expect AI transformation of public services. However, due to the early stage of the diffusion of generative AI technology, there are not many studies on how to implement AI systems in the public sector, and there is an urgent need to clarify the related concepts, considering that not only technical aspects, but also institutional aspects and social impact aspects should be considered together. Therefore, this study proposes a conceptual modelling of public AI systems considering the Korean context. For this purpose, we used the "conceptual framework development" method of Var-pio et al. (2020) to organise the attributes of 82 related literatures and concepts derived from the PRISMA guidelines, and conceptually derived a public AI system by connecting the relationships of each attribute. We distinguished four stages in the development of AI systems in the public sector - AI adoption, AI implementation, AI transformation, and AI evaluation - and defined the concept of a public AI system by linking the components of each stage, defining the attributes, and connecting the relationships between the factors. This can be used as a practical guide for the public sector, including the government, and will contribute to promoting the adoption of AI systems.