Artificial intelligence is becoming a defining force in public governance, yet many institutions still struggle to adopt it in ethical, sustainable, and scalable ways. This article reports on a systematic literature review in line with PRISMA 2020 guidelines, covering 67 peer-reviewed studies published between 2014 and 2024. The review shows that AI can help public institutions work faster and more transparently, but it also reveals several common problems. Many organizations still face fragmented data, weak connections between systems, limited digital tools, a lack of staff skills, and ethical risks such as bias and privacy concerns. To address these problems, the study introduces the AI Integration Capability Model, a framework based on the Technology Acceptance Model, Digital-Era Governance, and Dynamic Capabilities theory. The model highlights four institutional pillars: data access and interoperability, digital infrastructure and redesigned processes, workforce skills and learning capacity, and leadership and management reform. Its relevance was tested through a three-round Delphi study with 15 senior experts from Moroccan public institutions, who agreed on the feasibility and urgency of all four pillars. The findings offer policymakers practical guidance for AI adoption and outline a roadmap for aligning innovation with institutional readiness and public trust.