The Nosé-Hoover thermostat is a widely-used technique in molecular-dynamics simulations for maintaining a canonical ensemble at a desired temperature. The associated differential equation, known as the Nosé-Hoover equation, plays a crucial role in understanding the dynamics of the system. This thesis investigates the integrability of the Nosé-Hoover equations using the Darboux theory of integrability. It presents original proofs of the Darboux non-integrability of these equations (thereby identifying and correcting several major mistakes in the previous work by Adam Mahdi and Claudia Valls. “Integrability of the Nosé–Hoover equation”. In: Journal of Geometry and Physics 61.8 (2011), pp. 1348–1352.).